Codes from zero-divisors and units in group rings
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We study in this paper randomized constructions of binary linear codes that are invariant under the action of some group on the bits of the codewords. We study a non-Abelian randomized construction corresponding to the action of the dihedral group on a single copy of itself as well as a randomized Abelian construction based on the action of an Abelian group on a number of disjoint copies of itself. Cyclic codes have been extensively studied over the last 40 years. However, it is still an open question as to whether there exist asymptotically good binary cyclic codes. We argue that by using a slightly more complex group than a cyclic group, namely, the dihedral group, the existence of asymptotically good codes that are invariant under the action of the group on itself can be guaranteed. In particular, we show that, for infinitely many block lengths, a random ideal in the binary group algebra of the dihedral group is an asymptotically good rate-half code with a high probability. We argue also that a random code that is invariant under the action of an Abelian group G of odd order on k disjoint copies of itself satisfies the binary Gilbert-Varshamov (GV) bound with a high probability for rate 1/k under a condition on the family of groups. The underlying condition is in terms of the growth of the smallest dimension of a nontrivial F2-representation of the group and is satisfied by roughly most Abelian groups of odd order, and specifically by almost all cyclic groups of prime order.