Bounds on minimum distance for linear codes over GF(5)
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.
scientific article; zbMATH DE number 1340550