A Gilbert-Varshamov type bound for linear codes over Galois rings

In this paper we derive a Gilbert-Varshamov type bound for linear codes over Galois rings. For linear codes over the Galois ring GR(pl; j) the result can be stated as follows. Given r; such that 0 < r < 1 and 0 < H 1 pj (1 r): Then for all n N; where N is a sufficiently large integer, there exist [n; k] GR linear codes over GR(pl; j) such that k=n r and d=n : Consequently, this bound does not guarantee existence of better linear codes over GR(pl; j) than the usual Gilbert-Varshamov bound for linear codes over the residue class field GR(pj): 1 Linear codes over Galois rings A Galois ring R is defined to be a finite commutative local ring with unity, where the maximal ideal m is given by m = pR; p a prime number. The characteristic then is pl for some l 2 N ; and the residue class field is GF (pj) where j 2 N: Except for an isomorphism, R is uniquely determined by pl and j ([3], [2]), and we will let R = GR(pl; j) denote the Galois ring of characteristic pl and residue class field GF (pj): The set of n tuples over R; Rn; is a R module. By a linear code over R we mean any R submodule of Rn: We will denote such a linear code a GR linear code. Both authors are with the Department of Mathematics, Institute for Electronic Systems, Aalborg University DK-9220 Aalborg Ø, Denmark.