New [48,16,16] Optimal Linear Binary Block Code

ALinear binary [n, k]-code is a k-dimensional subspaces of the n-dimensional vector space GF (2) over the finite field GF (2) with 2 elements. The 2 codewords of length n are the elements of the subspace, they are written as row vectors. The weight wt(c) of a codeword is defined to be the number of nonzero entries of c and the minimum distance dist(C) of a code C is the minimum of all weights of the nonzero codewords in C. For the purpose of error correcting codes we are interested in codes with high minimum distance d as these allow the correction of ⌊(d − 1)/2⌋ errors. An [n, k]−code with minimum distance d is called an [n, k, d]−code. On the other hand we are interested in codes of small length n. High minimum distance and small length are contrary goals for the optimization of codes. There are several [5], [4] tables giving upper limits for the minimum distance of a linear code of fixed length n. A linear code C is called optimal if dist(C) is equal to this known upper bound. In a recent paper the authors first found by a computer program and later constructed an new optimal [47, 15, 16]−code. They also conjectured, that there is an [48, 16, 16]−code. As 16 is the maximum possible minimum distance of a code with parameters n = 48 and k = 16 such will be also optimal. In this paper we give a construction of such a code.