By Cunsheng Ding

This is often the 1st monograph on codebooks and linear codes from distinction units and nearly distinction units. It goals at delivering a survey of structures of distinction units and nearly 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 important mathematical foundations and the fundamentals of coding idea. It additionally comprises tables of most sensible 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 concept, or electronic communications.

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**Example text**

The following is proved in Whiteman (1962). 13. Define ν = (n 1 −1)(n 2 −1)/N 2 . Let symbols be the same as before. Then −1 = ςµ N/2 ς e/2 if ν is even, if ν is odd, where µ is some fixed integer such that 0 ≤ µ ≤ e − 1. Recall Whiteman’s cyclotomic classes Wi(N) of order N defined before. The cyclotomic numbers corresponding to these cyclotomic classes are defined by (i, j ) N = Wi(N) + 1 ∩ W (N) j for any pair of i and j with 0 ≤ i ≤ N − 1 and 0 ≤ j ≤ N − 1. The following lemma summarizes a number of properties of the cyclotomic numbers of order N [Whiteman (1962)].

Then the following are equivalent: • f (x) is planar; • f (x) is a two-to-one map, f (0) = 0 and f (x) = 0 for x = 0; July 11, 2014 5:59 Codes from Difference Sets - 9in x 6in b1830-ch01 Mathematical Foundations page 27 27 • there is a permutation polynomial g(x) over GF(q) such that f (x) = g(x 2 ) for all x ∈ GF(q). ) A list of planar monomials over finite fields is documented in the following theorem. 22. The function f (x) = x s from GF( pm ) to GF( pm ) is planar when • s = 2; or • s = p k + 1, where m/ gcd(m, k) is odd [Dembowski and Ostrom (1968)]; or • s = (3k + 1)/2, where p = 3, k is odd, and gcd(m, k) = 1 [Coulter and Matthews (1997)].

X n ) in GF(q)n is even-like provided that ni=1 x i = 0, and is odd-like otherwise. The weight of binary even-like vector must be even, and that of binary odd-like vector must be odd. For an [n, κ, d] code C over GF(q), we call the minimum weight of the even-like codewords, respectively the oddlike codewords, the minimum even-like weight, respectively the minimum odd-like page 34 July 11, 2014 5:59 Codes from Difference Sets - 9in x 6in b1830-ch02 page 35 Linear Codes over Finite Fields 35 weight, of the code.