On Rank and Kernel of the Homogeneous Gray image of linear codes over the Galois ring GR(۴,m)

Let R = GR(۴,m) be the Galois ring of characteristic ۴ and cardinality ۴m, where m is a naturanumber. Let C be a linear code of length n over R and Φ be the Homogeneous Gray map on R^n. TheHomogeneous Gray image of C, i.e., C = Φ(C) is a code (possibly nonlinear) of length N = ۲^mn overF۲^m. In this paper, we show that for any r ∈ {۲k۰ + k۱, . . . , min(n + k۰, ۲k۰ + k۱ +(k۰۲))}, there existsa R-linear code C of length N = ۲^mn and type {k۰, k۱} with rank(C) = r, and vice versa. We alsoestablish a similar result for kernel, i.e., for any k ∈ {k۰ + k۱, k۰ + k۱ + ۱, . . . , ۲k۰ + k۱ − ۲, ۲k۰ + k۱},there exists a R-linear code C of length N = ۲mn and type {k۰, k۱} with k۲m(C) = k, and vice versa.

کلیدواژه ها:

Galois ring ، Homogeneous Gray map ، Linear code ، Generator matrix ، Rank ، kernel