Ring Compute-And-Forward Over Block-Fading Channels

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C&F to models investigate channel we general paper, more with dealing opc ig a euti ihrcmuainrts becau which rates, more computation design, higher using in code example, result [15]. For can of in C&F. rings compact to space initiated advantages modules, the several and brings enlarges rings extension to This extended been work. future has as C&F MIMO leavi for while block- codes step diagonal), lattice over are algebraic modest MIMO matrices C&F degenerated channel a as where for viewed take channels be codes may we (which lattice channels paper, fading algebraic this C&F proposing for In needed by channels. codi is a such structures reason, this algebraic over For bette more setting. perform no with antenna to is single technique [8] a CSI of in Without that coding gain than transmitters. random multiplexing at the by however, available provided precoding, that is in than CSI gain better multiplexing if is the showed C&F [8] MIMO multiple-hop (MAC), multip channel and (MIMO) multi-output access channels multi-input a relay In networks. two-way line divers for full where observed [10], was in proposed was lattices root-LDA on over codes lattice de- for (GM) coder, geometric-mean namely, and Also decoders, decoder diversity. (AM) integer-forcing arithmetic-mean time two derive of as [9] rates relay, interpreted the achievable at better antennas receive is multiple assuming suc which blocks several time, of consists in it that in different slightly over codes integers lattice using rational investigated were channels fading utiliz may scheme performance. coding multip improve a that to is which diversity, here crux offer The resources “block-fading”). term using for ob us-ttc hc en htte(adm fading (random) codewor the each of that duration the means over constant which stay coefficients quasi-static, be to sadsg aaee nprle needn aig h tw and the world, fading, physical independent if parallel equivalent the are in of models parameter properties design a by is dictated is fading hr aebe oewrsi h ieaueo C&F on literature the in works some been have There nqaisai aigcanl,tesrcueo & codes C&F of structure the channels, fading quasi-static In time-varying where [10] [9], are work our to related Closely hl h lc egh(rchrnetime) coherence (or length block the While Z ebr IEEE Member, e,tecanlmdli 9,[0 is [10] [9], in model channel the Yet, . T slreeog seas [12]–[14] also (see enough large is Z rcia & ceebased scheme C&F practical A . n ifrn resources different T nblock- in cessive the d Our sion the the le- ity ng lel ng er se d. le o e 1 r 2 the rational integers Z or Gaussian integers Z[i] may not integer coefficients, we resort to solving lattice problems over be the most suitable ring to quantize channel coefficients. It Z-lattices and provide a means to assure linear independency has been shown that using the Eisenstein integers Z[ω] [16], of multiple equations over K. [17] or rings from general quadratic number fields [18] can 3) We analyze the degrees-of-freedomO (DoF) of our pro- have better computation rates for complex channels. Since the posed coding scheme. The DoF of C&F over quasi-static lattice codes in these extensions are all K-modules ( K refers fading channels has been analyzed using the theory of Dio- to the ring of integers in number fieldOK), the messageO space phantine approximation in [25]–[27]. Our analysis of DoF for can also be defined over K due to the first isomorphism Ring C&F requires a new result of Diophantine approximation theorem of modules. O by conjugates of an algebraic integer. The original contribution Our goal in this paper is to explore the fundamental limits of our work is the proof of a Khintchin-type result for of C&F over block-fading channels by using algebraic lattices Diophantine approximation by conjugate algebraic integers built from number fields of degree n (n 2). In quasi-static (Lemma 4). It is well known that the standard Khintchine channels, the C&F protocol essentially≥ builds on capacity- and Dirichlet theorems [28] only deal with the approximation achieving lattice codes for the additive white Gaussian-noise of real numbers by rationals, which are algebraic numbers (AWGN) channel [19]. To perform C&F in block-fading of degree one. Although some results on approximating a channels, we employ universal lattice codes proposed in [13], real number by an algebraic number are available in literature [14] for compound block-fading channels. The celebrated [29], [30], these results come with various restrictions which Construction A has been extended to number fields in recent unfortunately do not lend themselves to our problem at hand. years [12], [13], [18], [20], [21]. In [12], the authors proposed For instance, [29] only addresses simultaneous approximation algebraic lattice codes based on Construction A over K so of one number by algebraic conjugates or multiple numbers by that the codes enjoy full diversity; subsequently it was proO ved non-conjugates of a bounded degree, while [30] requires the in [13], [14] that such generalized Construction A can achieve real numbers to be approximated lie in a field of transcendence the compound capacity of block-fading channels. It was also degree one. briefly suggested in [22] that number-field constructions as in The rest of this paper is organized as follows. In Section II, [13], [14], [18] could be advantageous for C&F in a block- we review some backgrounds on algebraic number theory and fading scenario. C&F. In Sections III and IV, we present our Ring C&F scheme In this work, we propose a scheme termed Ring C&F and analyze its computation rates, respectively. In Section V, based on such algebraic lattices. As an extension of [23], we we analyze the achievable DoF without CSI at transmitters. elaborate the construction of algebraic lattices for Ring C&F, Subsequently Section VI provides some simulation results. and provide a detailed analysis using the geometry of numbers The last section concludes this paper. and Diophantine approximation. The main contributions of this Notation: The sets of all rationals, integers, real and com- work are the following: plex numbers are denoted by Q, Z, R and C, respectively. log 1) We propose Ring C&F over block-fading channels based denotes logarithm with base 2, and log+(x) = max(log(x), 0). K on lattice ΛO ( ) from generalized Construction A, which Matrices and column vectors are denoted by uppercase and C T K T satisfies relation K /ΛO ( ) / K , where T is the number lowercase boldface letters, respectively. dg(x) represents a O T K C I T of channel uses, , ΛO ( ) and K denote lattices built matrix filling vector x in the diagonal entries and zeros in OK C I from ring K itself, code and ideal K, respectively. Such the others. The operation of stacking the columns of matrix O C I algebraic lattices are shown to be K-submodules so that X one below the other is denoted by vec (X). x denotes O they are multiplicatively closed. The relay aims to decode the Euclidean norm of vector x, while X kdenotesk the an algebraic-integer linear combination of lattice codewords, Frobenius norm of matrix X. denotes thek Kroneckerk tensor ⊗ which means that the channel coefficient vectors are quantized product, and denotes the finite field summation. Λ( ) to a lattice which is the canonical embedding of the ring of is the nearest⊕ neighbor quantizer to a lattice Λ. (Λ)Q ,· integers . As a comparison, the lattice partition in a real T V K x R Λ(x)= 0 denotes the fundamental Voronoi re- quasi-staticO channel is ZT /ΛZ ( ) / (pZ)T , in which p is a ∈ Q C ngion of lattice Λ. [X]o mod Λ denotes [vec (X)] mod Λ. prime number. Also note the difference from techniques in

[16], [17] where channel coefficients are quantized to complex II. PRELIMINARIES quadratic ring K itself. Since the channel coefficients in O different fading blocks are unequal with high probability, it We first introduce necessary backgrounds on number fields is advantageous to employ the canonical embedding of K so and lattices (readers are referred to texts [31]–[33] for an O as to enjoy better quantization performance. introduction to these subjects), then review the protocol of 2) We analyze the computation rates in Ring C&F based on C&F over quasi-static channels. the universal coding goodness and quantization goodness of algebraic lattices. The quantization goodness of algebraic lattices constructed from quadratic number fields [18] is extended to A. Number Fields and Lattices general number fields. The semi norm-ergodic metric in [24] is Definition 1 (Number field). Let θ be a complex number with adopted to handle the effective noise. Regarding the equivalent minimum polynomial mθ of degree n. A numberfield is a field block-fading channel, the universal lattice codes in [14] play extension K , F(θ) that defines the minimum field containing an important role. In order to determine optimal algebraic- the base field F and the primitive element θ. 3

A number c is called an algebraic integer if its minimal A real Z-lattice is a discrete Z-submodule of Rm. Such a m m polynomial mc has integer coefficients. The maximal order of lattice Λ′ generated by a basis D = [d1,..., dm] R × an algebraic number field is its ring of integers. Let S be the set can be written as a direct sum: ∈ of algebraic integers, then the ring of integers is = K S. K D Zd Zd Zd The set θ ,θ , ..., θ n is called an integralO basis∩ of Λ′( )= 1 + 2 + + m. { 1 2 n} ∈ OK ··· K if c K, c = c1θ1 + c2θ2 + ... + cnθn with ci Z. With canonical embedding σ, an K-module Λ of rank m can O ∀ ∈ O ∈ O An element u K is called a unit if it is invertible under be transformed into a Z-lattice , and we write . ∈ O Λ′ Λ′ = σ (Λ) multiplication. All the units of K form a multiplicative group If K is a totally real number field of degree n, then we have an O mn mn , referred to as the unit group. embedded basis D R × , and we define its discriminant U ∈ 2 An embedding of K into C is a homomorphism into C as discK = det(D) . The successive minima λ (Λ′) of the | | i that fixes elements in Q. For a number field of degree n, Z-lattice Λ′ are defined in the usual manner. Analogously, we there are in total n embeddings of K into C: σi : K C, may define successive minima of Λ over K. i = 1,...,n, referred to as canonical embedding. Canonical→ O embedding establishes a correspondence between an element Definition 4 (Successive minima of modules [34]). The ith of an algebraic number field of degree n and an n-dimensional successive minimum of an K-module Λ is the smallest real number r such that the ballO (0, r) contains the canonical vector in the Euclidean space. The embeddings of θ, denoted B by σ (θ) n , are determined by the roots of m . We denote embedding of i linearly independent vectors of σ (Λ) over K: { i }i=1 θ by r1 the number of embeddings with image in R and by 2r2 1 λi(Λ) = inf r dim spanK σ− (σ (Λ) (0, r)) i . the number of embeddings with image in C. The pair (r1, r2) ∩B ≥ is called the signature of K. In a totally real number field, n o Notice that λ1 (Λ) =λ1(Λ′), and in general λi(Λ) λi(Λ′) . ≥ (r1, r2) = (n, 0) for i> 1. Also, if x1,..., xm are linearly independent over K The following two quantities of an algebraic number are of and achieve the successive minima of Λ, then the embeddings particular interest: σ(x1),...,σ(xm) are linearly independent and primitive in , n 1) The trace of θ: Tr(θ) i=1 σi(θ) F; the Euclidean lattice Λ′. n ∈ m m 2) The norm of θ: Nr(θ) , σi(θ) F. For any real Z-lattice Λ′(D) with D R × , Minkowski’s P i=1 ∈ ∈ In this work, we are only concerned with the scenario of real first theorem states that [35] Q channels and hence totally real number fields, so we use F = 2 2 λ (Λ′) κ det(D) m , (1) Q as the base field. For an extension to complex channels, one 1 ≤ m| | can choose F = Q(i) as the base field. and Minkowski’s second theorem states that m Definition 2 (Ideals and prime ideals). Let R be a commuta- 2 m 2 λi (Λ′) κm det(D) , (2) tive ring with identity 1R =0. An ideal I of R is a nonempty ≤ | | i=1 subset of R that has the following6 two properties: Y 2 2/m where κ , sup ′ D λ (Λ ) / det(D) is called Her- 1) c1 + c2 I if c1, c2 I; m Λ ( ) 1 ′ ∈ ∈ mite’s constant. | | 2) c1c2 I if c1 I, c2 R. ∈ ∈ ∈ Analogous bounds exist for the successive minima of - An ideal p of R is prime if it has the following two properties: K module Λ. Obviously, O 1) If c1 and c2 are two elements of R such that their product 2 2 c c is an element of p, then either c p or c p; λ (Λ) κ det(D) mn , (3) 1 2 1 2 1 ≤ mn| | 2) p is not equal to R itself. ∈ ∈ since the first minimum is identical. Applying Minkowski’s Every ideal of R can be decomposed into a product of prime second theorem to [36, Theorem 2] yields ideals. In particular, if p is a rational prime, we have pR = m g ei 2n mn 2 i=1 pi in which ei is the ramification index of prime ideal λ (Λ) κ det(D) . (4) i ≤ mn| | pi. The inertial degree of pi is defined as fi = [R/pi : Z/pZ], i=1 Q g Y and it satisfies i=1 eifi = n. Each prime ideal pi is said to lie above p. B. C&F over Quasi-Static Fading Channels P Definition 3 (Modules). A R-module is a set M together with Consider an AWGN network with L source nodes which a binary operation under which M forms an Abelian group, cannot collaborate with each other and are noiselessly con- and an action of R on M which satisfies the same axioms as nected to a final destination. We assume that all source those for vector spaces. nodes are operating with the same message space W (over finite fields [7] or rings [15]), and the same message rate Let D be a subset of R-module M. D forms an R-module R = 1 log( W ). Let ΛZ, ΛZ be a pair of nested lattices basis of M if every element in M can be written as a finite mes T | | c f T Z Z T linear combination of the elements of D. The order of the basis in the partition chain Z/Λf /Λc / (pZ) , in which p is a is called the rank of the module. A finite subset d ,... ,d prime number growing with the lattice dimension. A message { 1 m} of distinct elements of M is said to be linearly independent wl W is mapped bijectively into a lattice code via xl = over R if whenever m c d = 0 for some c ,... ,c R, (w∈) γΛZ, satisfying a power constraint of x 2 T P . i=1 i i 1 m ∈ E l ∈ f k lk ≤ then c1 = = cm =0. γ denotes a parameter to control the transmission power, and ··· P 4

P denotes the signal power, hence the signal-to-noise ratio User 1 Relay 1 Z1 (SNR) if the noise variance is normalized. w1 wˆ 1 ε ⊗ D The noisy observation at a relay is 1 ⊗ ⊕ 1 ⊗ L ⊗ Inverter y x z (5) ⊗ Z2 = hl l + , ⊗ w2 wˆ 2 l=1 ε ⊗ D X 2 ⊗ ⊕ 2 L where the channel coefficients h = [h ,... ,h ]⊤ R , and 1 L ∈ the additive noise z (0, IT ). The relay aims to compute User 2 Relay 2 a finite field equation∼ N Fig. 1: Compute-and-Forward over block-fading channels with L 2 users and 2 relays. u = alwl (6) Ml=1 L This has subsequently been improved by Ordenlitch, Erez and with coefficient vector a = [a ,... ,a ]⊤ Z and forward 1 L ∈ Nazer [26] to u, a to the destination. Each u corresponds to a lattice equation 1 L Z dcomp = . l=1 alxl mod γΛc as they are isomorphic. By first L estimating the lattice equation and then map it to a finite hP i field, the forwarded message from the relay is written as III. RING C&F L uˆ = (y h, a). We say equation u = l=1 alwl is decoded In this work, we consider a block-fading scenario where withD probability| of error δ if Pr (u = uˆ) < δ. diversity is supplied in n blocks and fading coefficients remain 6 L constant in each frame of coherence time T . That is, the Definition 5 (Achievable Computation Rate for a Chosen a fading process experienced by a codeword x of user l consists at a Relay). For a given channel coefficient vector h and a l of n blocks h1,l,h1,l,...,h1,l , h2,l,h2,l,...,h2,l , , chosen coefficient vector a, the computation rate Rcomp (h, a) { } { } ··· is achievable at a relay if for any δ > 0 and T large enough, T T hn,l,hn,l,...,hn,l in parallel. Thus the received signal at there exist encoders 1,... L and decoders such that the { } E E D | {z } | {z } relay can recover its desired equation with error probability T a relay can be written in matrix form as bound δ if the underlying message rate Rmes satisfies: | {z } L Rmes < Rcomp (h, a) . Y = HlXl + Z, (9) l=1 Theorem 1 ( [7]). There is a sequence of nested lattice code- X n T books ΛZ, ΛZ of length T , such that by setting T , where Y R × , Hl = dg (h1,l,... ,hn,l) denotes the f c → ∞ ∈ X Rn T the following computation rate is achievable: channel coefficients from user l to the relay, l × n o denotes a transmitted codeword to be designed in the∈ sequel, n T 1 P and Z R × is the additive noise with entries drawn from h a + ∈ Rcomp ( , )= max log 2 . (7) (0, 1). The index of the relay is dropped in the equation for 2 α R α 2 + P αh a ! ∈ | | k − k simplicityN of notation. The C&F diagram for this model with Upon receiving L linearly independent equations in the form two users (source nodes) and two relays is shown in Fig. 1. In of (6), the destination estimates the messages by inverting the the figure, the encoded messages (w1) and (w2) are both equations. The maximum information rate that the destination transmitted by using two sub-channelsE in parallel,E which are can receive through the AWGN network is dictated by the respectively denoted by black and blue arrows. Relays 1 and 2 computation rates at the relays. forward two linearly independent equations to the destination which subsequently recovers message wˆ , wˆ by inverting the Definition 6 (Achievable Computation Rate of the AWGN 1 2 L L equations. Network). Given hl l=1, and al l=1 from L relays such { } L { } Next, we present our Ring C&F scheme, which contains that the morphism of a is invertible in the message l l=1 message encoding based on algebraic lattices (such that the space, the achievable computation{ } rate of the AWGN network degree of the number field equals to the number of blocks in is min R (h , a ). l comp l l the block-fading model), and decoding algebraic-integer linear To characterize the the growth of computation rate w.r.t. combinations of lattice codewords. The “goodness” properties SNR, define the DoF as of algebraic lattices are shown in the last subsection.

maxa Rcomp (h, a) dcomp = lim . (8) P 1 A. Encoding →∞ 2 log (1 + P ) We follow [12], [13], [18] to build lattices from Construc- Using the theory of Diophantine approximation, Niesen and tion A over number ﬁelds. Choose a prime ideal p lying above Whiting [25] showed that rational prime p with inertial degree f so that we have an 1 isomorphism K/p = F f . Let be a (T, k) linear code over 2 ,L = 2; ∼ p dcomp 2 O C ≤ , L> 2. Fpf where k 2.

1 3 2 with z′ = z Z . l 1 1 l ∈ C. Goodness of Algebraic Lattices Definition 7 (Moments). The second moment of a lattice 2 Example 2. Consider quadratic field K = Q √5 . Choose ´ Z x dx Z nT 2 Z V(Λ )k k 2 R , Z p = 5, so the ideal factorization becomes p K = p , where Λ is σ˜ Λ nT (Λ ) , and the normalized ⊆ |V | Z 5 √5 5+3√5 O σ˜2 Λ p = − Z + − Z. For the isomorphism Fp = K/p, Z Z ( ) 2 2 second moment ofΛ is G Λ , Z 2/(nT ) . 2 ∼ O (Λ ) the five coset representatives in R corresponding to F5 are |V | Definition 8 (Quantization goodness) . A sequence of lattices Z nT 1 √5 1+ √5 Λ R is called good for MSE quantization if [0, 0]⊤, [1, 1]⊤, [− − , − ]⊤, ⊆ 2 2 1 lim G ΛZ = . 1 √5 1+ √5 T 2πe [ − , ]⊤, [ 1, 1]⊤. →∞ 2 2 − − The existence of such lattices has been shown in [38]. For lattices built from Construction A over quadratic fields, the Let the two uncoded messages be w = 2 for User 1 and 1 quantization goodness has been proved in [18] following [24]. w = 3 for User 2. For γ = 1, the transmitted lattice points 2 In the following theorem, we extend the quantization goodness are to lattices constructed from general number fields, whose proof is given in Appendix A. 1 √5 1+ √5 X = (w ) = [− − , − ]⊤, 1 E 1 2 2 Theorem 2. There exist a sequence of lattices in the ensemble 1 √5 1+ √5 (12) which are good for MSE quantization. X = (w ) = [ − , ]⊤. 2 E 2 2 2 7

Definition 9 (Universal coding goodness). For a block-fading Remark 2. If we confine a ZL in the above ∈ L channel in the form of y = Hx + z, with channel H theorem, then obviously Rcomp Hl , a a Z Rn Z RnT ∈ { } ∈ ≤ dg ( ) IT , codeword x Λ , and noise z admitting L 2⊗ ∈ ∈ Rcomp Hl , a a K , namely, the rate achieved by Z- (0, σz InT ), define the generalized volume-to-noise ratio { } ∈ O N lattice codes of length nT can only be lower. (VNR) as 2 The above theorem leads to the computation rate of the Z nT det (H) Λ block-fading network, which is simply the minimum compu- µ HΛZ , |V | . 2 tation rate among relays while making the set of combi- σz L nation coefficients invertible. In the following, we focus on A sequence of lattices ΛZ RnT is called universally good understanding the computation rate at one relay, as well as its for coding if for any µ HΛ⊆Z > 2πe, the error probability of Z extension to the multiple access scenario. estimating x given H satisfies Pe(Λ , H) 0 for all H . → Evaluating the K coefficient vector a is crucial in under- Theorem 3 ( [13], [14]). There exist a sequence of lattices in standing the performanceO limit of the computation rate. Our the ensemble (12) which are universally good for coding in goal is to find one coefficient vector or multiple coefficient block-fading channels. vectors minimizing the so-called additive Humbert form [39] Coding over algebraic lattices and coding over Z-lattices n have some differences, which we highlight in the following. F (a)= σj (a)⊤Mjσj (a). (19) j=1 1) Relation to coding using a rank-nT Z-lattice. The alge- X Z braic lattice Λ is a special case of rank-nT Z-lattices. With Cholesky decomposition of the L L matrix Mj = Z Z n × 2 Its extraordinary feature is that dg(σ (al)) Λ Λ . M¯ ⊤M¯ j , we may write F (a) = M¯ j σj (a) . This × ⊂Z j j=1 It also has a constant lower bound on , K dmin(Λ ) induces a squared distance over an K-module ΛO M¯ j , n jT Z 2 OP ¯ minx Λ 0 j=1 t=(j 1)T +1 xt , so the lattice en- whose generator matrix is given by the tuple M j , and ∈ \ − joys full diversity in block fading channels [12]. On multiplication in the module is defined over the embedded Q P the contrary, for an arbitrary lattice constructed from space. a random Construction A over Z, e.g., Λ′, it may have Let a1,..., aL be the coefficient vectors of the L K- K ¯ O dmin(Λ′)=0. successive minima of ΛO Mj . Define the equation rate 2) Relation to coding using n rank-T Z-lattices. If we just w.r.t. the ith coefficient vector a as i transmit n short lattice codewords of length T , then we n n will lose diversity and coding gain. R ( H )= log+ . (20) achv,i { l} 2 F (a ) i IV. ACHIEVABLE COMPUTATION RATE We refer to R ( H ) as the optimized (in the sense achv,1 { l} The main results in this section are Theorems 4 and 5, of optimizing the coefficient vectors) computation rate, and L R ( H ) as the optimized computation sum-rate. whose proofs will be given in the subsections. We reemphasize i=1 achv,i { l} here that our results only require channel knowledge at the TheoremP 5. The optimized computation rate satisfies receivers, not at the transmitters. L We begin by defining a , [a1,... ,aL]⊤ K , hj , Rachv,1 ( Hl ) L ∈ O n { } ≥ [hj,1,... ,hj,L]⊤ R , and Hl as the shorthand notation 1 n κ L ∈ { } + 2 + nL 1/n H log 1+ P hj log (discK) ; of l l=1. The definitions of the achievable computation 2L k k − 2 n { } j=1 rates in one relay and the whole block-fading network are X the same as those in Definitions 5 and 6, except that the (21) channel coefficient here is Hl , and the coefficient vector and the optimized computation sum-rate satisfies: a is algebraic. { } L Theorem 4. With our coding scheme in block-fading channels, R ( H ) achv,i { l} ≥ the following computation rate for a chosen a at a relay is i=1 X achievable as T : n 1 + 2 nL + κnL 1/n → ∞ log 1+ P hj log (discK) . R ( H , a)= 2 k k − 2 n comp { l} j=1 X (22) n + nP max log ; (17) Remark 3. While Eq. (22) serves as a characterization of the 2 b  n b 2 + P b h σ (a) 2  j=1 | j | k j j − j k performance of the L best linearly independent combinations,   and by optimizingPb in (17), we have: our coding technique should be further generalized (for this equation) to allow for L fine lattices (one per user) as well as n + n a form of successive interference cancellation at the receiver Rcomp ( Hl , a)= log n , { } 2 j=1 σj (a)⊤Mj σj (a) ! in order to create effective channels that only involve the (18) subset of lattices that can tolerate the increased varying noise where M I P h h P. faced when decoding each linear combination. For quasi-static j = P h 2+1 j j⊤ − k j k 8 channels, such a scheme is developed by Ordentlich et al. in so we obtain [26]. Our generalization follows in the same manner. L Y , BY γ A D mod γΛZ, Remark 4. Theorem 5 resembles its quasi-static counterpart in eff − l l c [26, Theorem 3], [27, Theorem 6]. The sum-rate is understood l=1 L X in the context of block-fading MAC, whose sum capacity is = AlXl + l=1 n X 1 lattice codeword log+ 1+ P h 2 . 2 k j k L j=1 | 1{z } ˜ Z X Ea Ea− (BHl Al) Xl + BZ mod γΛc , (25) · − ! Xl=1 The theorem shows that, for any SNR, the computation rate effective noise Zeff and sum-rate are never much smaller than the symmetric in which| Ea = dg ([E1{z, E2,... ,En]) with} capacity and sum-capacity of block-fading MAC. Since the gaps are determined by n, L and discK, one should choose a b 2 + P b h σ (a) 2 | n| k n n − n k number field with the smallest possible discriminant. En = 1 , q n n b 2 + P b h σ (a) 2 j=1 | j | k j j − j k Q q 1 L and Zeff = E−a (BHl Al) X˜ l + BZ represents an A. Proof of Theorem 4 l=1 − effective noise. We then use the semi norm-ergodicity in [24] P to characterize Zeff . With dithering, the transmitted codeword is given by X˜ l = Z ˜ Definition 10 (Semi norm-ergodicity [24]). A random vector [Xl + γDl] mod γΛc . The signal vec Xl is then uni- Z x of length T is called semi norm-ergodic with effective formly distributed over and is statistically indepen- 2 γ Λc variance 1 E x if for any ǫ,δ > 0, and T large enough, dent of vec (X ) accordingV to the Crypto lemma [19, Lemma T k k l 1]. After MMSE scaling as well as removing the dithers, we 2 have Pr x / 0, (1 + δ) E x ǫ. ∈B k k !! ≤ r L In Appendix B, we show that: BY γ AlDl Lemma 3. The random vector Z is semi norm-ergodic − vec ( eff ) l=1 with effective variance L X L n 1 = BHlX˜ l + BZ γ AlDl 2 2 2 n − σeff , bj + P bj hj σj (a) . (26) l=1 l=1 | | k − k X X j=1 L L L Y 1 = A X + BH X˜ + BZ A (X + γD ) . The matrix Ea can be viewed as the channel matrix in l l l l − l l l − l=1 l=1 l=1 Definition 9. By inspection of the proof of Theorem 3 in [14], X X X (23) it is not difficult to see that Theorem 3 also holds for semi norm-ergodic noise, similarly to [24]. We omit the details. To proceed, we need the following lemma. Therefore, there exist a sequence of lattices in the ensemble (12) such that the decoding error probability vanishes as T → Lemma 2. If A = dg (σ1(a), ..., σn(a)) with a K and as long as the VNR n T ∈ O ∞ S R × , then 2 ∈ 1 Z nT det E−a Vol γΛf > 2πe. (27) Z Z Z σ2 [AS] mod γΛc = A [S] mod γΛc mod γΛc . eff (24) On the other hand, the quantization goodness in Theorem 2 implies Proof: Write S = X + S′, where X is the closest lattice P 1+ δ Z (28) S 2 < vector of in γΛc . Then clearly both sides of Eq. (24) equal Z nT 2πe Z Z Vol (γΛc ) [AS′] mod γΛc , because Λc is also multiplicatively closed, similarly to Lemma 1. for any δ > 0 if T is large enough. It follows from (27) and Thus, the last term of Eq. (23) satisfies (28) that any computation rate up to Z 1 Vol(γΛc ) n P log Z < log (29) L L T Vol(γΛ ) 2 σ2 f ! eff A (X + γD ) mod γΛZ = A X˜ mod γΛZ, l l l c l l c is achievable. Xl=1 Xl=1 9

2 The effective noise variance σeff represents the geometric Second, if multiple message equations are required at one Z mean (GM) of the noise variances in all the blocks. The final relay, a search algorithm over Z-lattice Λ (ΦM¯ ) has to ensure 2 rate expression based on this form is given by : their coefficient vectors a1,..., aL are linearly independent over K. For the highest rates, it suffices to search for the K- n 1 O O R ( H , a)= log+ successive minima. This constraint can be incorporated into comp { l} 2 n 1/n j=1 (σj (a)⊤Mj σj (a)) ! an enumeration algorithm, which keeps increasing the search radius until linear independence is satisfied. The question that 1 + 1 Q = log n . (30) arises here is whether we can use the first few successive 2 σj (a) Mjσj (a) j=1 ⊤ ! minima of a Z-module to find those of an K-module. O Since the theQ optimization of the algebraic integer vector in Let a˜i be the vector giving the i-th successive minima Z 2 λi (ΦM¯ ) of Z-lattice Λ (ΦM¯ ). It may happen that a multiplicative form is complicated, we upper bound σeff by the arithmetic mean (AM) dim span K (Ψ ([a˜1,..., a˜L])) < L. n O 1 2 2 2 For example, choose K = Q √3 . Let a˜ = [1, 2, 1, 1]⊤, σAM , bj + P bj hj σj (a) 1 n | | k − k 2 j=1 a˜ = [6, 9, 4, 5]⊤; after mapping them back to , we have X 2 OK to reach (17), following (29). This enables the applications a1 = 1+ √3, 2+ √3 ⊤, a2 = 6+4√3, 9+5√3 ⊤. Since 3+ √3 a = a , one concludes that a and a are not of a nice algorithmic framework based on successive minima 1 2 1 2 in the next subsection. Lastly, the details of deriving (18) are independent over K. O given in Appendix C. Nevertheless, we have the following result: Proposition 2. Let the mapping Ψ be defined as in (31). Sup- B. Searching the Optimal Coefficients pose Z-coefficient vectors a˜1,..., añL produce the nL succes- Z In this subsection, we show that F (a) can be written as the sive minima λ1 (ΦM¯ ) ,... ,λnL (ΦM¯ ) of Z-lattice Λ (ΦM¯ ). squared distance of a Z-lattice vector, and explain the relation Then Ψ(a˜1),..., Ψ(añL) contains the L K-successive minima.{ } O between Z-successive minima and K-successive minima. These results enable the applicationO of conventional lattice Proof: Write the Z-coefficient matrix T = [a˜1,..., añL]. algorithms over Z to find one or multiple coefficient vectors nL nL From the definition of successive minima, T Z × is at a relay. We refer readers to [41]–[43] for these algorithms. ∈ a full-rank matrix such that ΦM¯ T = M¯ (Φ IL)T yields First, each K M¯ has a corresponding Z-lattice Z ⊗ ΛO j λ1 (ΦM¯ ) ,... ,λnL (ΦM¯ ) of Z-lattice Λ (ΦM¯ ). Notice that Z that belongs to a submodule of RnL, whose gen- Λ (ΦM¯ ) the L nL algebraic-integer matrix [a1,..., anL]=Ψ(T) erator matrix is × simply consists of the the first L rows of (Φ IL)T; in fact ΦM¯ = M¯ (Φ I ), ⊗ ⊗ L we have where [a ,..., a ] = [φ I ,...,φ I ] T. (32) M¯ 0 1 nL 1 L n L 1 ··· 0 0 Since φ1,...,φn is an integral basis of K, the matrix M¯ =  . ···. .  , [φ I ,...,φ{ I ] obviously} has rank L. ThenO it follows from . . . 1 L n L   the rank identity  0 M¯   ··· n    rank(C1C2) = rank(C1) and recall that Φ = [σ(φ ),...,σ(φ )] and φ ,...,φ is 1 n { 1 n} an integral basis of K. To show this more explicitly, note for full-rank matrix C2 that the matrix [a1,..., anL] is that there exists a bijectiveO mapping Ψ: ZnL L defined of rank L. Therefore, there exist exactly L vectors in → OK by a ,..., a which are linearly independent over K. Thus, { 1 nL} O the L K-successive minima must be contained in the set a = Ψ (a˜) O Ψ(a˜1),..., Ψ(añL) . n n n { } ⊤ The proposition shows searching for L K-independent O = φka˜(k 1)L+1, φka˜(k 1)L+2,..., φka˜kL ; lattice points inside ball (0, λ (ΦM¯ )) is possible. We − − nL "k=1 k=1 k=1 # further explain PropositionB 2 in Fig. 2. Suppose L = 3 and X X X (31) n = 3. There are 9 successive minima in the embedded since σj is a ring homomorphism, it follows that real lattice Λ(ΦM¯ ), and their corresponding algebraic coefficient vectors are denoted by a(1),1,..., a(3),3, where the n n ⊤ vectors in the same row are linearly dependent over K. The σj (a)= σj (φk)ã(k 1)L+1,..., σj (φk)ãkL . O − a(1),1, a(2),1, a(3),1 marked in red are coefficient vectors of "k=1 k=1 # X X the first three successive minima over K. 2 nL O Thus, F (a) = ΦM¯ a˜ (a˜ Z ) represents the squared k kZ ∈ distance of a point in Λ (ΦM¯ ). C. Proof of Theorem 5

2 n a ⊤M a To derive the optimized computation rate and sum-rate, we Here, j=1 σj ( ) j σj ( ) is called a multiplicative Humbert form [40]. Q only need to apply Minkowski’s ﬁrst and second theorems 10

F (a) increasing Finally, after substituting (35) into (18), we have:

Independence L a a a Rachv,i ( Hl ) (1),1 (1),2 (1),3 { } i=1 a a a X L (2),1 (2),2 (2),3 n n a a a = log+ (3),1 (3),2 (3),3 2 λ2 Λ K M¯ i=1 i O j ! X 1/n L n h 2 Fig. 2: Illustration of K-successive minima for L = 3 and n n j=1 1+ P j O log+ k k n =3. Among the 9 successive minima of the embedded real  L L/n  ≥ 2 Q κ (discK) lattice, those marked in red are coefficient vectors of the first nL n   three successive minima over K.  nL 1 + 2 1 + κnL L O = log 1+ P h log (discK) . 2 k j k − 2 nnL j=1 X K constant to ΛO M¯ j . First, by applying Sylvester’s determinant MAC capacity identity to each det(M¯ ) , one has | {z (36)} | i | | {z } n n 1/2 V. DOFANALYSIS det M¯ = det(M¯ ) = 1+ P h 2 − . | | | i | k j k j=1 j=1 Define DoF associated with Rachv,i as Y Y Rachv,i Consequently the volume of K M¯ becomes dachv,i = lim . (37) ΛO j P 1 →∞ 2 log (1 + P ) det(ΦM¯ ) = det(M¯ ) det(Φ I ) The main result of this section is: | | | || ⊗ L | n 1/2 L/2 2 − Theorem 6. For almost all Hl w.r.t. the Lebesgue measure, = (discK) 1+ P h . { } k j k the DoF’s of the optimized computation rate and sum-rate are j=1 n L Y respectively dachv,1 = L and i=1 dachv,i = n. Z The shortest lattice vector of Λ (ΦM¯ ) is the embedding Proof of Theorem 6: As aP direct consequence of Theorem K of the shortest lattice vector from ΛO M¯ j . Then it 5, the lower bounds of DoF’s are: follows from Minkowski’s first theorem over Z-lattices that L 2 K ¯ 2/(nL) n λ1 ΛO Mj κnL det(ΦM¯ ) , which yields d , d n. ≤ | | achv,1 ≥ L achv,i ≥ i=1 n X 1/(nL) n 2 K ¯ 1/n 2 − We will show in Theorem 7 that d , which is due λ1 ΛO Mj κnL (discK) 1+ P hj . achv,1 L ≤ k k to Lemma 4 on Diophantine approximation≤ of a real vector j=1 Y (33) by algebraic conjugates. The block-fading MAC capacity can upper bound the sum DoF’s, which yields L d n. i=1 achv,i ≤ By substituting (33) into the rate expression (18), we obtain Consequently, along with dachv,1 dachv,2 dachv,L, we have ≥ P ≥ n R d = = d = . achv,1 achv,1 ··· achv,L L n n = log+ 2 K ¯ 2 λ1 ΛO Mj ! Theorem 7. For almost all Hl w.r.t. the Lebesgue measure, 1/(nL) { } n 2 the DoF associated to the first computation rate satisfies n 1+ P hj n + j=1 n log k k dachv,1 .  1/n  ≤ L ≥ 2 Q κnL (discK) Lemma 4. Let ψ : N R+ be an approximation function. n   → 1  n κ Then for almost all Hl w.r.t. the Lebesgue measure, and for + h 2 + nL 1/n { } = log 1+ P j log (discK) . all q K, there exists a constant c′ H > 0 such that 2L k k − 2 n l j=1 ∈ O { } X constant 1 max min Hl dg (σ(a/q)) c′ H ψ( Nr(q) ) MAC capacity l 1,...,L a K l L × k − k≥ { } | | | {z (34)} ∈{ } ∈O (38) | {z } if ∞ ψ(k)nLkL < . k=1 ∞ Meanwhile, from Minkowski’s second theorem (4), we have PLemma 4 generalizes the classical Khintchine-Groshev theorem from Z to K. The proof is given in Appendix D. Note L O 2n K nL 2 that the approximation function in (38) can decay as fast as M¯ 1+L λj ΛO j κnL det(ΦM¯ ) . (35) +δ ≤ | | ( nL ) j=1 ψ( Nr(q) ) = Nr(q) − for any δ > 0. Lemma 4 Y | | | | 11

1 also indicates that, all points in the set q have the same where h , min˜ 2 . Thus, for almost all channel l∗ hl H˜ l ˜2 U ∈ (hl +1) approximation-error bound c′ H ψ( Nr(q) ). l | | realizations it holds that We proceed to prove Theorem{ } 7, where the technique is to B [H ,..., H ] [dg (σ (a )) ,..., dg (σ (a ))] 2 generalize the approach in [26], [27] to vectors of algebraic k 1 L − 1 L k 2 conjugates. ˜ max hl∗ Hldg (σ(q)) dg (σ (al)) Proof of Theorem 7: First rewrite the denominator ≥ l 2,... ,L − ∈{ } σ2 in (17) explicitly as a trade-off between “range” and 2 AM 2 ˜ 1 max hl∗ min σi(q) Hl dg(σ(q)) − dg (σ (al)) “accuracy”: ≥ l 2,... ,L i | | − ∈{ } 2 2 2 − c′′H min σi(q) Nr(q) − n − n(L 1) (42) 1 2 P 2 l B + B [H ,..., H ] [dg (σ (a )) ,..., dg (σ (a ))] ≥. { } i | | | | nk k n k 1 L − 1 L k where the last inequality is due to Lemma 4, and c′′ range accuracy Hl (39) depends on the realizations of H . { } { l} Let| {z0} stand| for the Voronoi region{z of 0 in the embedded } To analyze the “range” term of (39), we specify the gap 2 V mini σi(q) lattice σ( K). In the shortest vector problem (SVP), one aims among the embeddings of q: ̺ , | | 2 . Then the O maxi σi(q) to find a shortest nonzero vector, so the coefficients cannot | 2 | 2 analysis follows that of [26]. Since BH1 σ(q) /4 be σ (a1) = = σ (aL) = 0. By rearranging the order k k ≥ k k ··· if BH1 / 0, the first term of (39) satisfies of a1,... ,aL if necessary, we can assume that σ (a1) = 0. ∈V 6 n 2 Then the analysis falls into two cases depending on whether 1 2 1 ̺ maxi σi(q) B 2 2 bih1,i | 2| . BH1 0. n k k ≥ n max h | | ≥ 4max h ∈V i | 1,i| i=1 i | 1,i| i) If BH 0, then BH dg (σ (a )) is lower X (43) 1 ∈ V k 1 − 1 k bounded by the packing radius of lattice σ( K), which is Hereby we substitute (42) and (43) into (39): λ ( K) O 1 O . Based on this, we have 2 ̺c′′H P 2 2 ̺ 2 l { } L−1 σAM 2 max σi(q) + max σi(q) − ≥ 4maxi h1,i i | | n i | | 2 1 2 P 2 | | σ B + BH dg (σ (a )) 2 AM 1 1 2 L−1 ≥ n k k n k − k ρmin∗ max σi(q) + P max σi(q) − 2 2 ≥ i | | i | | L−1 λ1 ( K) λ1 ( K) L−1 1 > P > P L , (40) L L O O 1 L−1 1 − L−1 4n 4n ρ∗ P L + P L ≥ min L 1 L 1 − − ! where the first inequality is from (44)

′′ 2 ̺c H B [H1,..., HL] [dg (σ (a1)) ,..., dg (σ (aL))] ̺ l where ρ∗ , min 2 , { } , and the last in- k − k min 4 maxi h1,i n 2 | | BH1 dg (σ (a1)) . 2 ≥k − k equality follows from deﬁning x , maxi σi(q) and noticing 2 | | 2 that the convex function f (x) , x + P x− L−1 attains its − ii) If BH1 / 0, we have BH1 = dg (σ(q)+ ϕ) for 0 = L 1 ∈ V 6 P 2L σ(q) σ( K), ϕ 0. The “accuracy” term for two vectors minimum at root x = L 1 . ∈ O ∈V − H1 and Hl satisﬁes Finally, the lower bounds (40) and (44) on noise variance 2 2 L−1 σAM in both cases admit the inequality of σAM c′′′P L 2 ≥ B [H1, Hl] [dg (σ (a1)) , dg (σ (al))] for some constant c′′′. Substitute this lower bound on noise k − k 2 into the rate expression (17) and the DoF expression (37), one 2 ˜ ϕ + Hldg(σ(q)+ ϕ) dg (σ (al)) , (41) can show that d n . ≥k k − achv,1 ≤ L

˜ 1 where Hl = H1− Hl. The r.h.s. of (41) is a quadratic function VI. NUMERICAL RESULTS of ϕ. To attain its minimum, we solve the following equation In this section, we present numerical results to evaluate the performance of Ring C&F. Notice that there are many number 2 2 ﬁelds [44], [45] available to construct lattice codes: ∂ ϕ + H˜ ldg(σ(q)+ ϕ) dg (σ (al)) /∂ϕ = 0 k k − i) For relatively small n, we can enumerate all totally real number ﬁelds with small discriminants. Tables I to IV

˜ 2 1 ˜ ˜ 2 in Appendix E present this enumeration from quadratic to to get ϕ = (I+Hl )− Hlσ (al) Hl σ(q) . Substitute this − quintic number fields. According to the principle of small back into (41), we have discriminants shown in Theorem 5, the highest computation B H H 2 rates should come from quadratic to quintic number fields with [ 1, l] [dg (σ (a1)) , dg (σ (al))] 2 3 2 k − k 2 minimal polynomials mθ = θ θ 1, mθ = θ +θ 2θ 1, 1 4 3 2 − − 5 4 3 2− − H˜ 2 + I − H˜ dg (σ(q)) dg (σ (a )) mθ = θ +θ 3θ θ+1 and mθ = θ +θ 4θ 3θ +3θ+1, ≥ l n l − l − − − − respectively. 2 ˜ ii) For relatively large n, we can use the maximal real sub- hl∗ Hldg (σ(q)) dg (σ (al)) , ≥ − field of a cyclotomic number field. A cyclotomic field Q (ζk)

12 18

16 10 14

8 12

10 6 8

4 6 Ergodic rate/bpcu Ergodic rate/bpcu 4 2 2

0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 SNR/dB SNR/dB (a) Quadratic ﬁelds, n = 2. (b) Cubic ﬁelds, n = 3.

25 30

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20 15 15 10 10 Ergodic rate/bpcu Ergodic rate/bpcu

5 5

0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 SNR/dB SNR/dB (c) Quartic ﬁelds, n = 4. (d) Quintic ﬁelds, n = 5.

70 90

80 60 70 50 60

40 50

30 40 30 Ergodic rate/bpcu 20 Ergodic rate/bpcu 20 10 10

0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 SNR/dB SNR/dB (e) Cyclotomic fields, n = 11. (f) Cyclotomic fields, n = 14. Fig. 3: The ergodic computation rates (dashed lines) and sum-rates (solid lines) based on number fields of different degrees. 13

is a number field obtained by adjoining ζk to Q, where ζk rate of joint maximum likelihood (ML) decoding is represents a primitive kth root of unity. Its degree is n = n 1 ˆ ˆ ϕ (k) /2, where ϕ ( ) is Euler’s totient function. Table V in min log det I + P Hj, Hj,⊤ · 1,...,L 2n S S Appendix E shows the properties of maximal real sub-fields S⊂{ } |S| j=1 1 X Q ζk + ζk− with degrees n = 11, 14. with Hˆ j, being a submatrix consists of the columns of Hˆ j. In Fig. 3, we compare the optimized computation rate As shownS in Fig. 4, unlike the C&F setting,S IF based on Z and sum-rate of ring C&F and classic C&F, in terms still has full DoF, thanks to the cooperation among all receive E H of ergodic rate metrics defined as (Rachv,1 ( l )) and antennas. However, IF using Q √3 , Q √2 , and Q √5 in L { } E i=1 Rachv,i ( Hl ) . The expectation is taken over 2 Fig. 4-(a) provides approximately 4 5dB gain compared to { } × 3 Monte Carlo runs, with channel coefficients admitting that based on Z. The gain rises to around− 8dB in Fig. 4-(b) 10P (0, 1) entries. The “Z” curve in Fig. 3 denotes the classic for a large block size of n = 11. Thus, similarly to C&F, the N C&F using length-nT Z-lattice codes. The “mθ” curves, e.g., ring structure offers significant gains in IF. θ2 θ 1, denote ring C&F based on field Q (θ). For − − 1 cyclotomic number fields, we mark them with Q ζk + ζk− . VII. CONCLUSIONS The simulation starts by choosing L =2, n =2 in Fig. 3-(a), The class of algebraic lattices for C&F proposed in this then repeats by choosing n = 3, 4, 5, 11, 14 in Fig. 3-(b) to paper are built from Construction A over number fields. These Fig. 3-(f). Simulations can be made for the setting of larger L lattices enjoy the advantage of closure under multiplication in the same manner. by algebraic integers. Since the embeddings of an algebraic In Fig. 3-(a), significant performance gains can be observed integer are different, it helps to quantize block fading channels for Ring C&F. The quadratic field with minimal polynomial in a finer manner. Their achievable rates outperform those of 2 mθ = θ θ 1 performs superiorly to all other quadratic Z-lattices. − − 2 fields, and its sum-rate is within 1dB gap to the MAC capacity. Although we relaxed the GM σeff in (29) to the AM so that The DoF’s of Ring C&F for computation rates and sum-rates the problem was reduced to finding the successive minima are respectively 1 and 2. The classic C&F using Z gives very of a lattice, an important open question is how to minimize 2 poor rates. It falls behind Ring C&F with mθ = θ 3 by more the GM (a product form) efficiently, and how to analyze its than 25dB and increasing SNR results in little performance− Diophantine approximation. gain. Similar observations can be made from Fig. 3-(b) to Metric Diophantine approximation associated with K- Fig. 3-(f). They confirm that fields with minimal polynomials modules studied in this paper is more involved thanO that 3 2 4 3 2 5 mθ = θ +θ 2θ 1, mθ = θ +θ 3θ θ+1 and mθ = θ + associated with Z-lattices. We only addressed the convergent θ4 4θ3 3θ−2 +3−θ +1 are indeed− the best− for n =3, 4, 5. As part of the Khintchine-Groshev theorem, while the divergent predicted− − by the parameters in Theorem 5, the gaps between part was not used. We leave the divergent part of Lemma 4 1 1 the computation sum-rates of Q ζ23 + ζ23− , Q ζ29 + ζ29− as another open problem. and MAC capacities are much larger than those of quadratic fields, but their optimality in DoF is preserved. We further APPENDIX A explain why the classic C&F has roughly 0 DoF. From the law PROOF OF QUANTIZATION GOODNESS n of large numbers, we have approximation j=1 a⊤Mj a Our proof follows the steps in [24] with some adjustments: n(nP +1 P ) ≈ − a a for relatively large n; thus increasing SNR P 3n/2 nP +1 ⊤ P i) The prime number p is chosen to grow as O T rather 3/2 T k does not improve the rate. than O T , to compensate for the factor p − in the nT k The Ring C&F scheme can be extended to integer-forcing volume of the coarse lattice, while it is p − in [24]. ii) We (IF) for time-varying channels [9]. Suppose the channel ex- count the number of lattice points inside a ball for a number L L T nT periences n successive blocks Hˆ ,..., Hˆ R × (i.e., L field lattice σ K rather than an integer lattice Z . 1 n ∈ O single-antenna transmitters and one receiver with L antennas) Let VnT be the volume of an nT -dimensional unit ball. over the duration of a codeword. We can use our algebraic Set the inertial degree f = 1, and the scaling factor γ = 1/n 1/(2n) 1 lattices to show that the following rate is achievable in IF: p− discK− √nT . Write (10) explicitly as ρ− ( ) = T C ( )+ p , where ( ) maps Fp onto the coset leaders Mof eachC /p basedM on component-wise· isomorphism. The ˆ K RIF Hl = scaled latticeO is n o 1 nP K T + γΛcO = γ (Gw)+ γp , max min log n , L×L M [a1,... ,aL] K l 1,...L 2 σj (al)⊤Fj σj (al) ∈O ∈{ } j=1 ! where the volume of its embedded lattice satisfies rank[a ,... ,a ]=L 1 L Z nT T/2 (T k) P Vol γΛc γ discK p − and the equality holds only ≥ T k if the generator matrix G Fp × of has full rank. 1/2 ∈ Z C 1 ˆ ˆ − Since obviously limT G γΛc 1/ (2πe), we are left in which Fj = P − I + Hj⊤Hj . The difference from →∞ ≥ L L with the task of showing that that for any δ > 0, ǫ> 0, [9, Theorem 1] is that al rather than Z . Again, we ∈ OK compare the ring-based IF and Z-based IF in terms of ergodic σ˜2 γΛZ 1 Pr c > + δ <ǫ (45) rate E RIF Hˆ l . The channel capacity which equals the Z 2/(nT ) 2πe Vol (γΛ c ) ! n o 14

6 6

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2 2 Ergodic rate/bpcu Ergodic rate/bpcu

1 1

0 0 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 SNR/dB SNR/dB (a) Quadratic fields, n = 2. (b) Cyclotomic field, n = 11. Fig. 4: Ergodic rates of IF receivers with channel variation based on number fields of different degrees. with large enough T . Letting 0 <α< log (nT ), and ST = pT . With 0 <ρ<α, for any fixed x, the probability of| a| small quantization distance is bounded as nT 2/(nT ) α k , log V − 2 , (46) 2 log (p) nT Z ρ ε , Pr d x,γΛc 2− we have ≤ 1 2 ρ = Pr min (x γ (c))∗ 2− c (G) nT − M ≤ Z 2/(nT ) 2/(nT ) α ∈C Vol γΛ = nT V 2− . (47) c nT T T ρ = p− γS ∗ x, √nT 2− ∩B Denote by r the covering radius of the embedded lattice T T ρ o = p− γσ K x, √nT 2 T T T/2 O ∩B − σ K . Since Vol σ K = discK , the number of O 1/(2n) O T T 1/(2n) T 1 1/(2n)√ ρ points of discK− σ K inside a ball can be measured = p− discK− σ K x,γ − discK− nT 2− O nT O ∩B with volumes. Then we can adapt [24, Lemma 1] from Z (a) nT T T 1 1/(2n) ρ 1/(2n) to σ to get the following lemma. VnT p− γ− discK− √nT 2 discK− ro OK ≥ − − nT (b) nT/2 Lemma 5. For any x R and r> 0, the number of points k 2/(nT ) α − ρ nT/2 nT ∈ 1/(2n) T VnT p− nT VnT 2− nT 2− (1 γro) of a scaled lattice discK− σ inside (x, r) can be ≥ − OK B bounded as (c) ρnT/ 2 = VnT 2− O (1) , (48)

1/(2n) T discK− σ K (x, r) O ∩B T/2 1/(2n) where (a) is from Lemma 5, (b) is from using discK− Vol x, r disc K− r o . 1 ≥ (T k) nT Z − ≥ B − p − γ Vol γΛc and Eq. (47), and (c) has used Eq. nT (46) and (1 γro) = O (1). To see this, notice that Assume the source x is uniformly distributed over a fun- − r = O √nT as it is upper bounded by the length of damental region of lattice σ pT . For a target x RnT , its o a corner point of a Gram-Schmidt parallelepiped [43, Eq. distance to the closest lattice point equals to that modulo∈ the (44)], and that is independent of . If we choose coarse lattice: discK T p to grow with T cn, c > 1, e.g., p = ξT 3n/2 and minimize

1 2 ξ [1, 2) under the constraint that p is a prime [24], then Z nT d x,γΛc = min x γ (c) γλ ∈ c G T nT 1/(2n) 1/n ( ),λ σ(p ) nT k − M − k (1 γr ) = 1 disc− p− O (nT ) = O (1) ∈C ∈ − o − K 1 2 w.r.t. T . = min (x γ (c))∗ , c (G) nT − M k k ∈C For the p 1 non-zero random wi Fp, deﬁne the indicator function− ∈ 2 2 , T Z γ rp in which ( )∗ ( ) mod σ p . Clearly, d x,γΛc nT , · · ≤ T where rp denotes the covering radius of ideal lattice σ p . 2 ρ Note that (c) is uniformly distributed over the coset leaders 1, if (x γ (Gwi))∗ 2− M T χi = − M 2 ≤ , ST of ( /p) as the elements of G are uniform over F , and ρ K p (0, if (x γ (Gwi))∗ > 2− O − M

which satisfies E (χi)= ε. From Chebyshev’s inequality, Proof: i) First, we show that zeff∗ admits density 0, σ2 I . As a linear combination of independent Gaus- pk 1 eff nT − N 1 Z ρ sian random variables, σeff− zeff∗ has a density Pr d x,γΛc > 2− Pr χi =0 ≤   1 1 i=1 fz∗ σ− Ea− (BH A ) I z ⊛ ⊛ X ˜1 eff 1 1 T k − ⊗ ···  1 p 1 1 1 1 1 − fz∗ σ− Ea− (BH A ) I z ⊛ fz σ− Ea− B I z Var pk 1 i=1 χi ˜L eff L L T eff T − − ⊗ ⊗ ≤ ε2 = (0, InT ) , (50) p P N < . where ⊛ refers to the convolution of density functions. Thus, (pk 1) ε − we obtain Together with Eqs. (46) and (48), one has 2 fz∗ (z)= 0, σ I . (51) eff eff nT nT N Z ρ 2 (α ρ+O(1)) Pr d x,γΛc > 2− < 2− − . (49) ii) Second, we upper bound the density of each dithered ˜ It follows from (49) that we can use the same arguments as variable Xl by that of z˜l∗. The constrained Voronoi region for in [24] to show the expected second moment is small. Finally, each X˜ is , γΛZ 0, (1 + δ) nT P , so the l Vl V c ∩B we complete the proof of (45) by using Markov’s inequality. ˜ density function of Xl becomes p

APPENDIX B 1/ l , if z l fx˜l (z)= |V | ∈V PROOF OF LEMMA 3 (0, otherwise. By the law of total probability, nT Z Also, V r γΛ (1 δ′) for any small δ′ > 0 as |Vl| ≥ nT eff c − 2 vec X˜ is semi-norm ergodic. Let b˜ be a random vector Pr vec (Zeff ) / 0, (1 + δ) nT σeff = l ∈B Z q uniformly distributed over a ball of volume Vol γΛc ; its 2 Pr ( = 1)Pr vec (Z ) / 0, (1 + δ) nT σ =1 density function fb˜ (z) upper bounds fx˜ (z): T eff ∈B eff T l nT Z q x z V r γΛ f˜l ( ) nT eff c Z 0 2 = 1 δ′. +Pr( = 0)Pr vec ( eff ) / , (1 + δ) nT σeff =0 , z T ∈B T fb˜ ( ) l ≤ − q |V | where The Gaussian variable ˜z∗ with density f˜z∗ (z) has the same l l ˜ second moment as that of the fundamental Voronoi region 0, if x vec Xl , Z ∃ ∈ γΛc . Combining the above, we arrive at n o 2 V =  E s.t., x / 0, (1 + δ) x ,δ> 0 fx˜ (z) fx˜ (z) fb˜ (z) T  ∈B k k l l nT c(T )  r = < (1 δ′) e , (52)  fz∗ (z) f˜ (z) fz∗ (z) − 1, otherwise. ˜l b ˜l  nT c(T )  where fb˜ (z) /f˜z∗ (z) 0, we can make Pr ( = 0) ǫ by increasing l T ≤ iii) Finally, notice that the density of zeff is: T because vec X˜ l are all semi norm-ergodic. Then we 1 can confine our discussion to the case of = 1. This fx E (BH A ) I z ⊛ ⊛ n o T ˜1 a− 1 1 T constraint enables us to show the density of the effective noise 1 − ⊗ ··· 1 fx Ea− (BH A ) I z ⊛ fz Ea− B I z , is tightly upper bounded by that of a Gaussian vector with the ˜L L − L ⊗ T ⊗ T techniques in [19, Lemma 11], without proving the algebraic so combining this with the arguments in steps i) and ii) prove s lattices are good for covering. the proposition.

Since the Gaussian vector zeff∗ is semi norm-ergodic with Proposition 3. Assume =1. Let 2 T effective variance σeff , we have L 1 zeff = Ea− IT (BHl Al) IT vec X˜ l + B IT z . 2 ⊗ − ⊗ ⊗ Pr zeff∗ / 0, (1 + δ) nT σeff 0. l=1 ! ∈B → X q Then there exists an i.i.d. Gaussian vector Together with Proposition 3, we have 2 L Pr vec (Zeff ) / 0, (1 + δ) nT σ =1 0 1 ∈B eff T → z∗ = Ea− I (BH A ) I ˜z∗ + B I z eff ⊗ T l − l ⊗ T l ⊗ T and the proof is completed. l=1 ! p X with density fz∗ (z) = 0, σ2 I , σ2 = PPENDIX eff eff nT eff A C N 1 n 2 2 n DERIVATION OF EQ. (18) bj + P bjhj σj (a) , ˜z∗ (0, P InT ), j=1 | | k − k l ∼ N 2 such that the density of z is upper bounded as Note that σAM is a convex function of b. By assuming Q eff 2 a to be fixed, the minimum of σAM is reached by setting L Lc(T )nT fz (z) (1 δ′) e fz∗ (z) , ∂σ2 /∂b = 0. From this we have, for j =1, 2,...,n, eff ≤ − eff AM , 1 (nT ) 1 where c(T ) 2 log 2πeG Λ + nT , and δ′, c(T ) 0 P σj (a)⊤hj → bj = . (53) as T . P h 2 +1 → ∞ k j k 16

1 2 By plugging (53) into P σAM, we have for a ﬁxed q. Note that q,ψ = uq,ψ for any unit u , since A A ∈ U 1 2 σAM P min Hl dg (σ(a/(uq))) = min Hl dg (σ(a/q)) a K k − k a K k − k n ∈O ∈O 1 2 2 2 (55) = b 1+ P h 2Pb σ (a)⊤h + P σ (a) nP j k j k − j j j k j k and since Nr(q) = Nr(uq) . This means that when investi- j=1 X gating a sequence| | of | q with| decreasing approximation error n 2 { } 1 P σ (a) h ψ( Nr(q) ), we only have to pick q modulo the unit group. = j ⊤ j 1+ P h 2 + | | nP  2 k j k  Denote by (c, r) a ball of radius r with centre at c. The j=1 P hj +1! B X k k Lebesgue measure of a ball of radius ψ( Nr(q) ) centred at n   | | 1 P σj (a)⊤hj 2 σ(a/q) is given by 2P σj (a)⊤hj + P σj (a) nP − 2 k k n/2 j=1 P hj +1! ! π X k k Υ ( (σ(a/q), ψ( Nr(q) ))) = ψ( Nr(q) )n. n B | | Γ( n + 1) | | 1 2 P 2 = σ (a) σ (a)⊤ h h⊤ σ (a) n k j k − j 2 j j j A congruence consideration shows that the number of points j=1 P hj +1 ! ! X k k σ(a/q) K in (54) is exactly Nr(q) . We further elaborate counting∈V latticeO points inside the| fundamental| Voronoi region

n K in Fig. 5. Then the total measure of q,ψ is bounded by 1  P  VO A = σj (a)⊤ I 2 hj hj⊤ σj (a). L n  − P hj +1  Υ ( q,ψ) (Υ( (σ(a/q), ψ( Nr(q) ))) Nr(q) ) j=1  k k  A ≤ B | | | | X n/2 L  ,M   j  π n   = ψ( Nr(q) ) Nr(q) . (56) Γ( n + 1) | | | | Then the computation| rate in{z (17) can be} written as 2 Further deﬁne

∞ ∞ n + 1 ψ , lim sup q,ψ = q,ψ, (57) Rcomp ( Hl , a)= log n , W A A { } 2 (1/n) σ (a) M σ (a) Nr(q) N=1 k=N q: Nr(q) =k j=1 j ⊤ j j ! | |→∞ \ [ | [ | PL as the subset of Hl for which (54) holds for infinitely many in which the free parameter is a K. { } ∈ O q modulo the unit group. Let q = (q) denote the principal ideal generated by q. Since APPENDIX D (q) = (qu) for any unit u , the set of algebraic integers DIOPHANTINE APPROXIMATION BY ALGEBRAIC can be partitioned into different∈ U subsets indicated by principle CONJUGATES ideals. Since Nr(q) = Nr(q) , the number of subsets q | | U Our proof may be viewed as an extension of Khintchine’s whose elements have absolute norm k is equal to the number theorem for complex numbers given in [46, Section 4], which of principal ideals with norm k. Consequently, we have, for dealt with Diophantine approximation by ratios of Gaussian N =1, 2, , , ··· ∞ integers. We first recall a result from [47, Theorem 5], [48, p. ∞ 132] to count the number of principal ideals in K. Υ O  Aq,ψ k=N q: Nr(q) =k Lemma 6. Let J(k, K) be the number of principal ideals in [ | [ |   L K with norm no larger than k. Then ∞ πn/2 O n ψ(Nr(q))Nr(q) ρK n n−1 n ≤ Γ( + 1) J(k, K) ρKk 2 k n max (1, Φ ) , k=N q: Nr(q)=k 2 | − |≤ w 0 X X n/2 L r r ∞ n 1 2n n rM(n 1) 2 1 (2π) 2 RK π nL L where Φ0 = 2 − n γ¯ e − , ρK = , w = n ψ(k) k 1 w√ discK Γ( + 1) | | 2 k=N q: Nr(q)=k denotes the number of roots of unity in K, RK denotes the X X regulator of the log-unit lattice, (r1, r2) is the signature of K, where q : Nr(q)= k denotes a principal ideal with norm k. By n−1 n−1 r = r1 + r2 1, and γ,¯ M are parameters of the log-unit Lemma 6, we have J(k, K)= ρKk+O k n . As O k n lattice. − grows no faster than k, we have q: Nr(q)=k 1= O (1), which Proof of Lemma 4: Firstly assume that H belongs to is bounded by a constant in the limit of k. l P K , the fundamental Voronoi region of lattice σ ( K), for By the Borel–Cantelli lemma [28], the Lebesgue measure VO O 1 l L. Let ties on the boundary of K be broken in an Υ ( ψ)=0 if Υ k∞=N q: Nr(q) =k q,ψ < . Obvi- O ≤ ≤ V W | | A nL ∞L arbitrary manner. Define ously, the convergence of the series ∞ implies S S k=1 ψ(k) k that Υ ( ψ)=0. Since a countably infinite number of the q,ψ , W A Voronoi regions cover the whole space,P our result holds for all H H Hl . In fact, it is readily verified that the set ψ is periodic l max min l dg (σ(a/q)) < ψ( Nr(q) ) { } W { } l 1,...,L a K k − k | | with respect to lattice σ( ). This establishes an algebraic ∈{ } ∈O K (54) version of Khintchin’s theoremO [28] in the convergent part.

(a) q = (5+ √5)/2, ψ( Nr(q) ) = 5−1.2. (b) q =4+ √5, ψ( Nr(q) ) = 11−1.2. | | | | 2 1+√5 Fig. 5: Approximating [h ,h ]⊤ R with K = Z[ ]. The well approximable set 1 2 ∈ O 2 [h1,h2]⊤ mina K [h1,h2]⊤ σ(a/q) < ψ( Nr(q) ) is shaded in blue. Black dots denote lattice points in σ ( K). ∈O − | | O Orangen dots a K σ(a/q) denote the centers of balls in approximatingo [h1,h2]⊤. ∪ ∈O

Since for almost all Hl , (54) holds for ﬁnitely many q [5] S. H. Lim, Y. Kim, A. E. Gamal, and S. Chung, “Noisy network coding,” { } IEEE Trans. Inf. Theory modulo the unit group, there exists a ﬁnite constant c H such , vol. 57, no. 5, pp. 3132–3152, May 2011. { l} [6] Y. Song and N. Devroye, “Lattice codes for the Gaussian relay chan- that nel: Decode-and-forward and compress-and-forward,” IEEE Trans. Inf. Theory, vol. 59, no. 8, pp. 4927–4948, Aug. 2013. max min Hl σ(a/q) ψ( Nr(q) ) [7] B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing interfer- l 1,...,L a K k − k≥ | | ∈{ } ∈O ence through structured codes,” IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6463–6486, Oct. 2011. for all Nr(q) c H . So one can claim that | |≥ { l} [8] J. Zhan, U. Erez, M. Gastpar, and B. Nazer, “MIMO compute-and- forward,” in Proc. IEEE Int. Symp. Inf. Theory, ISIT 2009, Seoul, Korea. max min H σ(a/q) c′ H ψ( Nr(q) ) l l IEEE, 2009, pp. 2848–2852. l 1,...,L a K k − k≥ { } | | ∈{ } ∈O [9] I. E. Bakoury and B. Nazer, “The impact of channel variation on integer- for all algebraic integer q with forcing receivers,” in Proc. IEEE Int. Symp. Inf. Theory, ISIT 2015, Hong Kong, China. IEEE, 2015, pp. 576–580. c′ H = [10] P. Wang, Y. Huang, K. R. Narayanan, and J. J. Boutros, “Physical- { l} layer network-coding over block fading channels with root-LDA lattice codes,” in Proc. IEEE Int. Conf. Commun., ICC 2016, Kuala Lumpur, maxl 1,...,L mina K Hl σ(a/q) min 1, min ∈{ } ∈O k − k . Malaysia. IEEE, 2016, pp. 1–6. q: Nr(q)

TABLE I: Real quadratic ﬁelds with small discriminants.

m θ2 θ 1 θ2 2 θ2 3 θ2 θ 3 θ2 θ 4 θ − − − − − − − − discK 5 8 12 13 17 basis φ 1, θ 1, θ 1, θ 1, θ 1, θ { } { } { } { } { } TABLE II: Real cubic ﬁelds with small discriminants.

m θ3 + θ2 2θ 1 θ3 3θ 1 θ3 + θ2 3θ 1 θ3 θ2 4θ 1 θ − − − − − − − − − discK 49 81 148 169 basis φ 1, θ, θ2 1, θ, θ2 1, θ, θ2 1, θ, θ2 TABLE III: Real quartic ﬁelds with small discriminants.

m θ4 + θ3 3θ2 θ + 1 θ4 + θ3 4θ2 4θ + 1 θ − − − − discK 725 1125 basis φ 1, θ, 1+ θ + θ2, 1 2θ + θ2 + θ3 1, θ, 2+ θ2, 1 3θ + θ3 − − − − − − m θ4 4θ3 + 8θ 1 θ4 4θ2 + θ + 1 θ − − − discK 1600 1957 basis φ 1, θ, 1 1 2θ + θ2 , 1 3 θ 3θ2 + θ3 1, θ, 2+ θ2, 1 3θ + θ3 2 − − 2 − − − − TABLE IV: Real quintic ﬁelds with small discriminants.

m θ5 + θ4 4θ3 3θ2 + 3θ + 1 θ5 5θ3 θ2 + 3θ + 1 θ − − − − discK 14641 24217 basis φ 1, θ, 2+ θ2, 3θ + θ3, 1 2θ 3θ2 + θ3 + θ4 1, θ, 2+ θ2, 1 4θ + θ3, 2 5θ2 + θ4 − − − − − − − − m θ5 + θ4 5θ3 3θ2 + 2θ + 1 θ5 + θ4 5θ3 θ2 + 4θ 1 θ − − − − − discK 36497 38569 basis φ 1, θ, 2+ θ2, 2 4θ + θ2 + θ3, 1 + 2θ 5θ2 + θ4 1, θ, 2+ θ + θ2, 3θ + θ2 + θ3, 3 2θ 5θ2 + θ3 + θ4 − − − − − − − −

TABLE V: Maximal real sub-ﬁelds of cyclotomic ﬁelds Q (ζk).

− − − Q 1 Q 1 Q 1 ζk + ζk ζ23 + ζ23 ζ29 + ζ29 φ (k) /2 11 14 11 10 9 8 7 6 14 13 12 11 10 9 8 − mζ +ζ 1 ζ + ζ 10ζ 9ζ + 36ζ + 28ζ ζ + ζ 13ζ 12ζ + 66ζ + 55ζ 165ζ k k − − − − − 56ζ5 35ζ4 + 35ζ3 + 15ζ2 6ζ 1 120ζ7 + 210ζ6 + 126ζ5 126ζ4 56ζ3 + 28ζ2 + 7ζ 1 − − − − − − − − disc − 41426511213649 10260628712958602189 Q ζ +ζ 1 k k

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Shanxiang Lyu received the B.Eng. and M.Eng. degrees in electronic and information engineering from South China University of Technology, Guangzhou, China, in 2011 and 2014, respectively, and the Ph.D. degree from the Electrical and Electronic Engineering Department, Imperial College London, in 2018. He is currently a lecturer with the College of Cyber Security, Jinan University. His research interests are in lattice theory, algebraic number theory, and their applications. [45] L. C. Washington, Introduction to Cyclotomic Fields. Springer-Verlag New York, 1996.

Antonio Campello received the Bachelor and PhD degrees in Applied Mathematics from the University of Campinas, Brazil, in 2009 and 2014, respectively. He was a visiting researcher at the Complutense University of Madrid in 2009, at the École Polytechnique fédérale de Lausanne (EPFL) 1025 in 2011, and at AT&T Research Labs - Shannon Labs, New Jersey [46] M. M. Dodson and S. Kristensen, “Hausdorff dimension and Diophan- in 2013. He was as a postdoctoral researcher at Télécom ParisTech, France, tine approximation,” ArXiv Mathematics e-prints, May 2003. in and at Imperial College London, UK. His research interests are in the interplay between discrete geometry, number theory, communications and machine learning.

Cong Ling (S’99-A’01-M’04) received the B.S. and M.S. degrees in electrical engineering from the Nanjing Institute of Communications Engineering, Nan- [47] M. R. Murty and J. V. Order, “Counting integral ideals in a number jing, China, in 1995 and 1997, respectively, and the Ph.D. degree in electrical ﬁeld,” Expositiones Mathematicae, vol. 25, no. 1, pp. 53–66, Feb. 2007. engineering from the Nanyang Technological University, Singapore, in 2005. He had been on the faculties of the Nanjing Institute of Communications Engineering and King’s College. He is currently a Reader (Associate Pro- fessor) with the Electrical and Electronic Engineering Department, Imperial College London. His research interests are coding, information theory, and security, with a focus on lattices. Dr. Ling has served as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS and the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. [48] S. Lang, Algebraic Number Theory. Springer-Verlag New York, 1994.