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

Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field F_p^m. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring R_p,m=F_p^m[u,v]/〈u²=v²=0,uv=vu〉, where m is a positive integer, α=a+ub+vc+uvd is a unit of R_p,m, a,b,c,d∈F_p^m, and a is nonzero. We then find a Gray map from R_p,m[x]/〈xⁿ−α〉 (with respect to homogeneous weights) to F_p^m[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 α-constacyclic codes over R_p,m of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over F_p^m 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.