Double quadratic residue codes and self-dual double cyclic codes

In this paper, we introduce double Quadratic Residue Codes (QRC) of length n = p + q for prime numbers p and q in the ambient space F-2(p) x F-2(q) . We give the structure of separable and non-separable double QRC over this alphabet and we show that interesting double QR codes in this space exist only in the case when p = q. We give the main properties for these codes such as their idempotent generators and their duals. We relate these codes to codes over rings and show how they can be used to construct interesting lattices. As an applications of these codes, we provide examples of self-dual, formally self-dual and optimal double QRC. We also provide examples of best known quantum codes that are derived from double-QRC in this setting.