Constructions of A Large Class of Optimum Linear Codes for Small Number of Information Symbols

摘要

A new method of constructing efficient codes on the basis of (it, u + v) construction is presented. We present a large class of optimum linear codes for small number of information symbols. Namely we present (3(2~m - 1), m + 1,3 ·2~m - 2) code, (3(2~m - 1) + 1, m + 1,3 ·2~m - 1) code and (3(2~m - 1) + 2, m + 1,3 ·2~m) code respectively. We show that these codes meet the bound of linear codes (m ^ 6) due to Brouwer and Verfoeff. It is strongly conjectured that, for m > 6, our codes meet the bound for any code-length. We then present the (n,2,d) codes that meet the bound of any linear code of length n ≤ 125. It is strongly conjectured that the proposed (n, 2, d) codes meet the bound of optimum linear code of any code length. We also present the (n, 3,d) code that meet the bound of any linear code of length n except n = 8 + 7μ for n ≤ 60. It is strongly conjectured that our (n, 3, d) code meet the bound for any code length except m = 8 + 7μ(μ= 1,2, ···. Finally we present several optimum linear codes for the number of information symbols k = 4,5,6 and 7.