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arXiv:2301.00753 · Advances in Mathematics of Communications

Polynomial representation of additive cyclic codes and new quantum codes

Reza Dastbasteh, Khalil Shivji

Published Oct 11, 2023• 7 citations
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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} $.

DOI
10.3934/amc.2023036

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QUATERNARY CONSTRUCTION OF QUANTUM CODES FROM CYCLIC CODES OVER F4+uF4\mathbb{F}_4 + u\mathbb{F}_4F4​+uF4​
In this paper, we establish the necessary and sufficient conditions for quasi-cyclic (QC) codes with index even to be symplectic self-orthogonal. Subsequently, we present the lower and upper bounds on the minimum symplectic distances of a class of 1-generator QC codes and their symplectic dual codes by decomposing code spaces. As an application, we construct many new binary symplectic self-orthogonal QC codes with excellent parameters, leading to 117 record-breaking quantum error-correction codes.