Bounds on minimum distance for linear codes over GF(5) (Q1304997)

Let \(GF (q)\) denote the Galois field of \(q\) elements, and let \(V(n,q)\) denote the vector space of all ordered \(n\)-tuples over \(GF (q)\). A linear code \(C\) of length \(n\) and dimension \(k\) over \(GF (q)\) is a \(k\)-dimensional subspace of \(V(n,q)\). Such a code is called an \([n,k,d;q]\)-code if its minimum Hamming distance is \(d\). Then a central problem in coding theory is that of optimizing one of the parameters \(n\), \(k\) and \(d\) for given values of the other two. Two versions are: Problem 1: Find \(d_q (n,k)\), the largest value of \(d\) for which there exists an \([n,k,d;q]\)-code. Problem 2: Find \(n_q (k,d)\), the smallest value of \(n\) for which there exists an \([n,k,d;q]\)-code. For the case of linear codes over \(GF(5)\), Problem 2 has been considered by \textit{R. Hill} [Optimal linear codes, in: Mitchel, C., (ed.) Cryptography and coding II, Oxford Univ. Press, 75-104 (1992; Zbl 0742.94012)] and solved for \(k<3\). In addition, \(n_5 (4,d)\) has been solved for all but 22 values of \(d\) by \textit{I. Boukliev, S. Kapralov, T. Maruta} and \textit{M. Fukui} [Optimal linear codes of dimension 4 over \(GF(5)\), IEEE Trans. Inf. Theory 43, 308-313 (1997; Zbl 0871.94046)]. Here the authors study the next four dimensions. Forty four new linear codes over \(GF(5)\) are constructed and a table of \(d_5(n,k)\), \(k\leq 8\), \(n\leq 100\) is presented.