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Linear Programming Bounds for Tree Codes (Corresp.) Linear Programming Bounds Peter Boyvalenkov [email protected] Institute for Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria; Technical Faculty, Southwestern University, Blagoevgrad, Bulgaria Danyo Danev [email protected] Department of Electrical Engineering and Department of Mathematics, Link¨oping University, SE-581 83 Link¨oping, Sweden Abstract This chapter is written for the forthcoming book ”A Concise Encyclopedia of Coding The- ory” (CRC press), edited by W. Cary Huffman, Jon-Lark Kim, and Patrick Sol´e. This book will collect short but foundational articles, emphasizing definitions, examples, exhaustive refer- ences, and basic facts. The target audience of the Encyclopedia is upper level undergraduates and graduate students. 1 Preliminaries – Krawtchouk polynomials, codes, and designs Definition 1.1 (Krawtchouk polynomials). Let n 1 and q 2 be integers. The Krawtchouk ≥ ≥ polynomials are defined as i (n,q) j i j j n j z K (z) := ( 1) (q 1) − q − , i = 0, 1, 2,..., i − − n i j Xj=0 − where z := z(z 1) (z j + 1)/j!, z R. j − · · · − ∈ (n,q) Theorem 1.2 ([57]). The polynomials Ki (z) satisfy the three-term recurrence relation (n,q) (n,q) (n,q) arXiv:1903.02255v1 [cs.IT] 6 Mar 2019 (i + 1)Ki+1 (z) = [i + (q 1)(n i) qz]Ki (z) (q 1)(n i + 1)Ki 1 (z) − − − − − − − with initial conditions K(n,q)(z) = 1 and K(n,q)(z)= n(q 1) qz. 0 1 − − Theorem 1.3 ([57]). The discrete measure n n n i dµ (t) := q− (q 1) δ(t i)dt, (1) n i − − Xi=0 where δ(i) is the Dirac-delta measure at i 0, 1,...,n , and the form ∈ { } f, g = f(t)g(t)dµ (t) (2) h i n Z define an inner product over the class of real polynomials of degree less than or equal to n. Pn 1 Theorem 1.4 (Orthogonality relations, [57]). Under the inner product (2) the Krawtchouk poly- nomials satisfy n n n K(n,q)(u)K(n,q)(u)(q 1)u = δ qn(q 1)i i j − u i,j − i Xu=0 for any i, j 0, 1,...,n . Moreover, ∈ { } n (n,q) (n,q) n Ki (u)Ku (j)= δi,jq . u=0 X n (n,q) Theorem 1.5 (Expansion, [57]). If f(z)= i=0 fiKi (z), then 1 n n P− n f = qn(q 1)i f(u)K(n,q)(u)(q 1)u i − i i − u u=0 n X n (n,q) = q− f(u)Ku (i). u=0 X n+1 Definition 1.6. Define K(n,q)(z) := q n (u z). Note that K(n,q)(z) is orthogonal to any n+1 (n+1)! u=0 − n+1 (n,q) polynomial Ki (z), i = 0, 1,...,n, with respectQ to the measure (1). Theorem 1.7 (Krein condition, [57]). For any i, j 0, 1,...,n ∈ { } n (n,q) (n,q) u (n,q) (n,q) Ki (z)Kj (z)= pi,jKu (z) (mod Kn+1 (z)) u=0 X with pu = 0 if i + j > u, pu > 0 if i + j = u n, and pu 0 otherwise. i,j i,j ≤ i,j ≥ Definition 1.8. Denote i 1 n − T n,q(z, w) := K(n,q)(z)K(n,q)(w) (q 1)j . i j j − j Xj=0 Theorem 1.9 (Christoffel-Darboux formula, [57]). For any i = 0, 1,...,n and any real z and w i + 1 (w z)T n,q(z, w)= K(n,q)(z)K(n,q)(w) K(n,q)(w)K(n,q)(z) . − i q(q 1)i n i+1 i − i+1 i − i Let F n be a code, where F = 0, 1,...,q 1 is the alphabet of q symbols (so q is not C ⊆ q q { − } necessarily a power of a prime). For x,y F n, recall that d(x,y) is the number of coordinates ∈ q where x and y disagree. Definition 1.10. The vector B( ) = (B ,B ,...,B ), where C 0 1 n 1 B = (x,y) 2 : d(x,y)= i , i = 0, 1,...,n, i { ∈C } |C| is called the distance distribution of . Clearly, B = 1 and B = 0 for i = 1, 2,...,d 1, C 0 i − where d is the minimum distance of . C 2 Definition 1.11. The vector B ( ) = (B ,B ,...,B ), where ′ C 0′ 1′ n′ n 1 (n,q) Bi′ = BjKi (j), i = 0, 1,...,n, |C| Xj=0 is called the dual distance distribution of or the MacWilliams transform of B( ). Ob- C C viously B0′ = 1. Theorem 1.12 ([36, 37]). The dual distance distribution of satisfies C B′ 0, i = 1, 2,...,n. i ≥ Theorem 1.13 ([60]). If q is a power of a prime and is a linear code in Fn, then B ( ) is the C q ′ C distance distribution of the dual code . C⊥ Definition 1.14. The smallest positive integer i such that B = 0 is called the dual distance i′ 6 of and is denoted by d = d ( ). Denote by s = s( ) (resp., s = s ( )) the number of nonzero C ′ ′ C C ′ ′ C B ’s (resp., B ’s), i 1, 2,...,n , i.e. i i′ ∈ { } s = i : B = 0,i> 0 , s′ = i : B′ = 0,i> 0 . |{ i 6 }| |{ i 6 }| The number s is called the external distance of . Define δ =0 if B = 0 and δ = 1 otherwise ′ C n (respectively, δ′ =0 if Bn′ = 0 and δ′ = 1 otherwise). Definition 1.15. Let F n be a code and M be a codeword matrix consisting of all vectors of C ⊆ q as rows. Then is called a τ-design if any set of τ-columns of M contains any τ-tuple of F τ C C q the same number of times (namely, λ := /qτ ). The largest positive integer τ such that is a |C| C τ-design is called the strength of and is denoted by τ( ). The number λ is called the index C C of . C n Remark 1.16. A τ-design in Fq is also called an orthogonal array of strength τ [37] or a τ-wise independent set [4] (see also [49]). Theorem 1.17 ([36, 37]). If F n has dual distance d = d ( ), then is a τ-design, where C ⊆ q ′ ′ C C τ = d 1. ′ − n (n,q) Definition 1.18. For a real polynomial f(z)= i=0 fiKi (z), the polynomial nP n/2 (n,q) f(z)= q− f(j)Kj (z) Xj=0 b n/2 is called the dual to f(z). Note that the dual to f(z) is f(z) and also that f(i)= q fi for any i 0, 1,...,n . ∈ { } b b 3 2 General linear programming theorems n Theorem 2.1 ([36, 37]). For any code Fq with distance distribution (B0,B1,...,Bn) and C ⊆ n (n,q) dual distance distribution (B0′ ,B1′ ,...,Bn′ ), and any real polynomial f(z) = i=0 fiKi (z), it is valid that n n P f(0) + Bif(i)= f0 + fiBi′ . |C| ! Xi=1 Xi=1 In Definition 1.9.1 in Chapter 1, Aq(n, d) is defined for codes over Fq. We now extend that definition to codes over Fq as only the alphabet size is important and not its structure. Definition 2.2. For fixed q, n, and d 1, 2,...,n denote ∈ { } A (n, d) := max : F n, d( )= d . q {|C| C ⊆ q C } Definition 2.3. For fixed q, n, and τ 1, 2,...,n denote ∈ { } B (n,τ) := min : F n, τ( )= τ . q {|C| C ⊆ q C } Theorem 2.4 (Linear Programming Bound for codes [36, 37]). Let the real polynomial f(z) = n (n,q) i=0 fiKi (z) satisfy the conditions (A1)P f > 0, f 0 for i = 1, 2,...,n; 0 i ≥ (A2) f(0) > 0, f(i) 0 for i = d, d + 1,...,n. ≤ Then A (n, d) f(0)/f . Equality holds for codes F n with d( ) = d, distance distribu- q ≤ 0 C ⊆ q C tion (B0,B1,...,Bn), dual distance distribution (B0′ ,B1′ ,...,Bn′ ) and polynomials f(z) such that Bif(i) = 0 and Bi′fi = 0 for every i = 1, 2,...,n. Remark 2.5. The polynomial f from Theorem 2.4 is implicit in Theorem 1.9.23 a) from Chapter n (n,q) 1 as i=0 BiKi (i) and it is normalized for f0 (or B0) to be equal to 1; note also the normal- ization of the Krawtchouk polynomials. Thus the bound f(1)/f appears as n B in Chapter P 0 i=0 i 1. P Theorem 2.6 (Linear Programming Bound for designs [36, 37]). Let the real polynomial f(z)= n (n,q) i=0 fiKi (z) satisfy the conditions (B1)P f > 0, f 0 for i = τ + 1,τ + 2,...,n; 0 i ≤ (B2) f(0) > 0, f(i) 0 for i = 1, 2,...,n. ≥ Then B (n,τ) f(0)/f . Equality holds for designs F n with τ( ) = τ, distance distribu- q ≥ 0 C ⊆ q C tion (B0,B1,...,Bn), dual distance distribution (B0′ ,B1′ ,...,Bn′ ) and polynomial f(z) such that Bif(i) = 0 and Bi′fi = 0 for every i = 1, 2,...,n. Remark 2.7. The Rao Bound in Theorem 3.3 and Levenshtein Bound in 3.10 below are obtained by suitable polynomials in Theorems 2.4 and 2.6, respectively.
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