Codes from Difference Sets by Cunsheng Ding

By Cunsheng Ding

This can be the 1st monograph on codebooks and linear codes from distinction units and nearly distinction units. It goals at supplying a survey of structures of distinction units and nearly distinction units in addition to an in-depth therapy of codebooks and linear codes from distinction units and nearly distinction units. To be self-contained, this monograph covers beneficial 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 one 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 conception, or electronic communications.

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It is easy to prove the following. (1) Let N ≡ 3 (mod 4). Then max1≤w≤N−1 |ACλ (w)| ≥ 1. On the other hand, max1≤w≤N−1 |ACλ (w)| = 1 iff ACλ (w) = −1 for all w ≡ 0 (mod N). In this case, the sequence λ∞ is said to have ideal autocorrelation and optimal autocorrelation. (2) Let N ≡ 1 (mod 4). There is some evidence that there is no binary sequence of period N > 13 with max1≤w≤N−1 |ACλ (w)| = 1 [Jungnickel and Pott (1999)]. It is then natural to consider the case max1≤w≤N−1 |ACλ (w)| = 3. In this case, ACλ (w) ∈ {1, −3} for all w ≡ 0 (mod N).

4]. 21 (Restricted Johnson Bound for A q (n, d, w)). 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 , .

2) Let N ≡ 1 (mod 4). There is some evidence that there is no binary sequence of period N > 13 with max1≤w≤N−1 |ACλ (w)| = 1 [Jungnickel and Pott (1999)]. It is then natural to consider the case max1≤w≤N−1 |ACλ (w)| = 3. In this case, ACλ (w) ∈ {1, −3} for all w ≡ 0 (mod N). (3) Let N ≡ 2 (mod 4). Then max1≤w≤N−1 |ACλ (w)| ≥ 2. On the other hand, max1≤w≤N−1 |ACλ (w)| = 2 iff ACλ (w) ∈ {2, −2} for all w ≡ 0 (mod N). In this case, the sequence λ∞ is said to have optimal autocorrelation. (4) Let N ≡ 0 (mod 4).

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