An infinite family of Griesmer quasi-cyclic self-orthogonal codes

Abstract

Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field F-pm. We first explicitly determine the generators of alpha-constacyclic codes over the finite Frobenius non-chain ring R-p,R-m = F-pm [u, v]/(u(2) = v(2) = 0, uv = vu), where m is a positive integer, alpha = a + ub + vc + uvd is a unit of R-p,R-m,R- a, b, c, d is an element of F-pm, and a is nonzero. We then find a Gray map from R-p,R-m[x]/(x(n) - alpha) (with respect to homogeneous weights) to F-pm [x]/(x(p3m+1n) - a) (with respect to Hamming weights), which is linear and preserves minimum weights. We present an efficient algorithm for finding the Gray images of alpha-constacyclic codes over R-p,R-m of length n, which produces infinitely many quasi-cyclic self orthogonal codes over F-pm of length p(3m+1) and index p(3m). In particular, some family turns out to be "Griesmer" codes; these Griesmer quasi-cyclic self-orthogonal codes are "new" codes compared with previously known Griesmer codes of dimension 4. (C) 2021 Published by Elsevier Inc.