Additive Polycyclic Codes over ${\mathbb{F}}_4$ Induced by Binary Vectors and Some Optimal Codes

Abstract

In this paper we study the structure and properties of additive right and left polycyclic codes induced by a binary vector $a$ in ${\mathbb{F}}_2^n.$ We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if $C$ is a right polycyclic code induced by a vector $a\in {\mathbb{F}}_2^n$, then the Hermitian dual of $C$ is a sequential code induced by $a.$ As an application of these codes, we present examples of additive right polycyclic codes over ${\mathbb{F}}_4$ with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over ${\mathbb{F}}_4.$