A Gilbert-Varshamov type bound for linear codes over galois rings
Bo Hove, C. Thommesen · 1998
In this paper we derive a Gilbert-Varshamov type bound for linear codes over Galois rings. For linear codes over the Galois ring GR(p l ; j) the result can be stated as follows. Given r; ffi such that 0 ! r ! 1 and 0 ffi ! H \\Gamma1 p j (1 \\Gamma r): Then for all n N; where N is a sufficiently large integer, there exist [n; k] GR linear codes over GR(p l ; j) such that k=n r and d=n ffi: Consequently, this bound does not guarantee existence of better linear codes over GR(p l ; j) than the usual Gilbert-Varshamov bound for linear codes over the residue class field GR(p j ): 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 p l for some l 2 N ; and the residue class field is GF (p j ) where j 2 N: Except for an isomorphism, R is uniquely determined by p l and j ([3], [2]), and we will let R = GR(p l ; j) denote the ...