Codes from Difference Sets by Cunsheng Ding

By Cunsheng Ding

This is often the 1st monograph on codebooks and linear codes from distinction units and virtually distinction units. It goals at offering a survey of structures of distinction units and virtually distinction units in addition to an in-depth remedy of codebooks and linear codes from distinction units and nearly distinction units. To be self-contained, this monograph covers helpful mathematical foundations and the fundamentals of coding thought. It additionally includes tables of top BCH codes and most sensible cyclic codes over GF(2) and GF(3) as much as size a hundred twenty five and seventy nine, respectively. This repository of tables can be utilized to benchmark newly built cyclic codes. This monograph is meant to be a reference for postgraduates and researchers who paintings on combinatorics, or coding thought, or electronic communications.

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A q (n, d, w) ≤ qw2 n(q − 1)d , − 2(q − 1)nw + n(q − 1)d provided that qw2 − 2(q − 1)nw + n(q − 1)d > 0. This bound is restricted because of the condition qw2 − 2(q − 1)nw + n(q −1)d > 0. 6]. 22 (Unrestricted Johnson Bound for Aq (n, d, w)). If 2w ≥ d and d ∈ {2e − 1, 2e}, then Aq (n, d, w) ≤ n qˆ w (n − 1)qˆ w−1 ··· (n − w + e)qˆ ··· w−1 , where qˆ = q − 1. 4 Cyclic Codes Over GF(q) An [n, κ] code C over GF(q) is called cyclic if c = (c0 , c1 , . . , cn−1 ) ∈ C implies (cn−1 , c0 , c1 , . . , cn−2 ) ∈ C.

Let C1 and C2 be two linear codes of the same length over GF(q). Then C1 and C2 are monomially equivalent if there is a monomial matrix over GF(q) such that C2 = C1 M. Monomial equivalence and permutation equivalence are precisely the same for binary codes. If C1 and C2 are monomially equivalent, then they have the same weight distribution. The set of monomial matrices that map C to itself forms the group MAut(C), which is called the monomial automorphism group of C. Clearly, we have PAut(C) ⊆ MAut(C).

X m ), defines a vector ( f (x))x∈GF(2)m ∈ GF(2)2 , where x ranges over July 11, 2014 5:59 Codes from Difference Sets - 9in x 6in b1830-ch02 Linear Codes over Finite Fields page 39 39 all elements in GF(2)m in some fixed order. This vector is called the truth table of f . It is easy to prove that any function f from GF(2)m to GF(2) can be expressed as a linear combination of the following 2m basis functions: Basis functions degree total number m 0 m 1 m 2 1 0 xi , 1 ≤ i ≤ m 1 xi x j , 1 ≤ i < j ≤ m 2 ..

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