Polynomial representation of additive cyclic codes and new quantum codes

Abstract

We give a polynomial representation for additive cyclic codes over ${\mathbb{F}}_{p^2}$. This representation will be applied to uniquely present each additive cyclic code by at most two generator polynomials. We determine the generator polynomials of all different additive cyclic codes. A minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic codes over ${\mathbb{F}}_p$. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal additive cyclic codes over ${\mathbb{F}}_{p^2}$. Finally, we present ten record-breaking binary quantum codes after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive cyclic codes over ${\mathbb{F}}_4$.

arXiv comment17 pages, 2 Tables