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A Method to Find the Volume of a Sphere in the Lee Metric, and Its A method to find the volume of a sphere in the Lee metric, and its applications Sagnik Bhattacharya, Adrish Banerjee Department of Electrical Engineering, Indian Institute of Technology Kanpur Email: {sagnikb, adrish}@iitk.ac.in Abstract—We develop general techniques to bound the size of B. Prior work the balls of a given radius r for q-ary discrete metrics, using Given a bound on the volume of the q-ary Lee ball of radius t, the generating function for the metric and Sanov’s theorem, that V (n), for a code of blocklength n, Chiang and Wolf [1] give reduces to the known bound in the case of the Hamming metric t and gives us a new bound in the case of the Lee metric. We use the Hamming bound the techniques developed to find Hamming, Elias-Bassalygo and (n) n(1−R(d)) Gilbert-Varshamov bounds for the Lee metric. V(d−1)=2 ≤ q (1) I.I NTRODUCTION and the Gilbert-Varshamov bound, (n) n(1−R(d)) A. Background Vd > q (2) One of the outcomes of Shannon’s classic work on information on the rate R(d) for a code of minimum distance d in the Lee theory is a long and ongoing investigation of point-to-point metric. They also give the following bound analogous to the communication over a discrete memoryless channel and the singleton bound for a linear code of length n and rank k in limits of reliable communication over such a channel. To the Lee metric, describe these limits, we can use various metrics defined on ( q+1 the codeword space that are matched to the channel under con- 4 (n − k + 1) for odd q d ≤ 2 (3) sideration, where we say that a metric is matched to a channel q 4(q−1) (n − k + 1) for even q if nearest neighbour decoding according to the metric implies maximum likelihood (ML) decoding on the channel [1]. The and the following method of calculating the volume of a sphere kind of metric chosen can vary, for example, depending on the in the Lee metric r ! kind of modulation scheme used for communication over the X 1 di V (n)(z) = A(n)(z) (4) channel. The limits also change depending on what kind of r i! dzi error criterion is being used - for example, whether we want i=0 z=0 average error or maximum error to be bounded. which is mathematically involved. Here A(n)(z) is the gener- The most well studied metric is the Hamming metric, which ating function for the Lee metric. is suited to orthogonal modulation schemes [2]. For the error Wyner and Graham [13] give the following Plotkin-type rate to be exactly zero, we have several upper bounds like nD bound for the Lee metric: d ≤ 1−|Cj−1 where the low-rate average-distance Plotkin bound [3], the Hamming ( 2 bound based on sphere packing [2], the Elias-Bassalygo bound q −1 for odd q D = 4q (5) [4] and the linear programming based MRRW bound [5], and q for even q lower bounds like the Gilbert-Varshamov bound [6] that tell 4 us what rates are possible and what are not, given an error and jCj is the size of the code. Berlekamp [2] used a result due criterion that the code is to meet, even though the question to Chernoff [14] to find the volume of a ball in the Hamming of what is the precise capacity is still open. Many of these metric using the generating function as a starting point, but bounds use at some point a bound on the number of codewords omits the corresponding calculations for the Lee metric as they of length n of a certain q-ary Hamming weight r, which is are too tedious. He also gives a version of the Elias-Bassalygo the size of the Hamming ball of radius r, and the standard bound for 0 < t < Dn: known result is that this size is qnHq (r=n) to first order in the t t d ≤ 2 − (6) exponent, where Hq(·) is the q-ary entropy function. 1 − K−1 nD There are other metrics of practical and mathematical inter- (n) Vt est, for example the Lee metric, introduced in [7], [8]. This where K is the least integer not less than qn(1−R) , and t is metric is known to be suited to phase modulation schemes [2], chosen to minimize the RHS of the inequality. and has more recently been used in multi-dimensional burst- Golomb [15] gave several results on spheres in several error correction [9], constrained and partial response channels different discrete metrics. [10], interleaving schemes [11], and error correction for flash Roth [16] also gives versions of the Hamming and Gilbert- memories [12]. Varshamov bounds for the Lee metric, and gives the following closed form expression for the volume of a Lee sphere of introduced in the previous section, we can write A(n)(z) = q t P (n) j (1) n radius t < 2 (here i = 0 if i > t). j Aj z = [A (z)] . Dividing both sides of the equation n n by q , we get that (n) X i n t Vt = 2 (7) (1) n (n) i i A (z) X Aj X (n) i=0 = zj = B zj (9) q qn j As an exercise, he also includes the result that his expression j j q is a strict lower bound for all t ≥ . (n) 2 (n) Aj C. Our contributions where Bj := qn Now, consider n i.i.d. random variables We develop a general technique based on the generating X1;:::;Xn, each distributed as the random variable X de- function of a metric and Sanov’s theorem to find the volume fined above. We have, for each k, of a sphere of a given radius. We show that this method 2 n 3 allows us to recover the familiar bounds on the volume of X (n) P 4 Xj = k5 = Bk (10) a Hamming ball. We find upper and lower bounds on the j=0 volume of a ball in the Lee metric, and we use this result P (n) to find bounds analogous to the Hamming, Elias-Bassalygo Therefore, to calculate a bound on the quantity j≤k Bj , P Pn and Gilbert-Varshamov bounds for the Lee metric. we need a bound on the quantity j≤k P [ i=0 Xi = j], D. Structure of the paper which we can calculate using Sanov’s theorem. Multiplying In section II, we introduce the necessary notation. In section III the bounds that we obtain by qn immediately gives bounds on (n) we describe the general method using Sanov’s theorem, and we Vt . The theorem states that [17] give specific examples of it’s use in section IV - the Hamming Theorem 1 (Sanov’s Theorem). Let X ;X ;:::;X be i.i.d. metric in section IV-A and the Lee metric in section IV-B. 1 2 n ∼ Q(x). Let E ⊆ P be a set of probability distributions and P Finally, in section V, we use the results developed in the be the set of all types from the n realisations X ;X ;:::;X . section IV-B to find Hamming and Gilbert-Varshamov bounds 1 2 n Then, for the Lee metric. ∗ Qn(E) = Qn(E \P ) ≤ (n + 1)jX j2−nD(P jjQ) (11) II.P ROBLEM SETUP AND NOTATION n We adopt a slightly modified version of the notation and where jX j is the support of each Xi, D(·||·) is the K- terminology of Berlekamp [2]. The discrete metric under L divergence, Qn(E) is the probability that the empirical consideration gives the distance between any two points in distribution obtained from an n-long sample X1;:::;Xn each the space of n-length vectors over an alphabet of q symbols. ∼ Q(x) belongs to the set E, and Given a center C and a radius r, define the sphere S(C; t) P ∗ = arg min D(P jjQ) (12) as the set of all points whose distance from C is less than or P 2E equal to t. The surface area of such a sphere is the number of is the distribution in E that is closest to Q in relative entropy. vectors whose distance from C is exactly t and it is denoted by If we also have that the set E is the closure of its interior, (n) At . The volume of such a sphere is the number of vectors then we also have the result (n) whose distance from C is ≤ t, and it is denoted as Vt . 1 (n) Pt (n) log Qn(E) ! −D(P ∗jjQ) (13) Clearly, we have the equality Vt = j=0 Aj . Now, let n (n) P (n) j (n) A (z) = j Aj z , the generating function for the Aj . Retaining terms upto first order in the exponent, we have Since the distance is additive over the n coordinates, the −nD(P ∗jjQ)−o(n) n −nD(P ∗jjQ)+o(n) generating function is multiplicative over these coordinates, 2 ≤ Q (E) ≤ 2 (14) (n) (1) n (1) and we have the equality A (z) = [A (z)] , where A (z) Suppose that we want the expression for the volume of a gives the weights for a single symbol only. For example, for ball of radius k, therefore we must first calculate the quantity the Hamming metric we have A(1)(z) = 1 + (q − 1)z, and for P Pn j≤k P [ i=0 Xi = j].
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