A Johnson-type bound for group codes and lattices

Abstract

In this work we give and analyze a Johnson-type bound for group codes considering the G-norm. Johnson bounds have been given for binary and q-ary codes [5, 7, 8] with respect to the Hamming distance. We borrow the idea of the G-norm from [3] and define a new distance for codewords: the G-semidistance. We extend the Johnson-type bounds for binary and q-ary codes to the Gsemidistance and give a relation between these bounds and our G-semidistance. By means of this, we present an upper bound on the number of codewords inside a G-ball and an l1-ball, within a certain given radius, for both group codes and lattices. Johnson-type bounds provide an upper bound on the number of codewords in a Hamming ball with a specified radius. The original proof is based on linear algebra [5, 8]; proofs with a geometric view are presented in [1]. The extension of Johnson-type bounds for q-ary codes is given in [7]. In all of these works the Johnson-type bounds use Hamming balls. Here we consider G-balls with an arbitrary received vector as the G-ball’s center, given a specified radius; we find an upper bound for the number of codewords in the G-ball. Roughly speaking, we investigate the number of codewords such that their G-semidistances from the received word is less than the radius of the G-ball. Recently the question of list decoding under the Hamming metric has become an important trend in coding theory. In addition, list decoding for lattices are given in [6, 9]. Our Johnson-type bound, when applied to some recent works [2, 3], may lead to list decoding of q-ary codes, group codes and lattices via the G-norm. Let x = (x1, . . . , xn) be in a group code G. The G-norm for x is defined as ‖x‖G = x1 + x2 + · · ·+ xn where the operations are performed in R [3].