Codes based on dyadic matrices and their generalizations

Meraiah Martinez; Tefjol Pllaha; Christine A. Kelley

doi:10.3934/amc.2024054

Article Contents

2025, Volume 19, Issue 5: 1277-1300. Doi: 10.3934/amc.2024054

This issue Previous Article Several infinite families of binary cyclic codes with square-root-like lower bounds Next Article Linkable ring signature scheme based on multivariate polynomial over finite field

Codes based on dyadic matrices and their generalizations

1.
Benedictine College, USA
2.
University of South Florida, USA
3.
University of Nebraska – Lincoln, USA

^*Corresponding author: Meraiah Martinez
^*Corresponding author: Meraiah Martinez

Received: February 2024

Revised: September 2024

Early access: December 13, 2024

Published online: December 13, 2024

The third author is partially supported by a Simons Collaboration Grant.

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Abstract

Codes based on algebraically structured parity check matrices are desirable in applications due to their compact representations, practical implementation advantages, and tractable decoder performance analysis. In this paper, we examine codes based on parity check matrices that are dyadic, $ n $-adic, or quasi-dyadic (QD), meaning the parity check matrix representation is block structured with dyadic matrices as blocks. Depending on the number of nonzero positions in the leading row of each matrix or block, these codes may be either low density or moderate density. We examine basic code properties of dyadic, $ n $-adic, and QD parity check codes, including reproducibility, bounds on the dimension and minimum distance, cycle structure of the corresponding Tanner graph, and their possible use in quantum code constructions.

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Mathematics Subject Classification: 94B05, 94B65.

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References

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