A family of cyclic codes with increasing dimensions and fixed minimum distances

Hanglong Zhang; Xiwang Cao; Gaojun Luo

doi:10.3934/amc.2026013

Article Contents

2026, Volume 23: 40-51. Doi: 10.3934/amc.2026013

This volume Previous Article Several classes of three-weight and four-weight linear codes Next Article Permutation polynomials over ${\mathbb{F}}_{{q}^{2}}$ constructed from self-conjugate reciprocal polynomials

A family of cyclic codes with increasing dimensions and fixed minimum distances

1.
Department of Basic Courses, Suzhou City University, Suzhou, Jiangsu 215104, China
2.
School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China
3.
Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, Jiangsu 211106, China

^*Corresponding author: Gaojun Luo
^*Corresponding author: Gaojun Luo

Received: December 15, 2024

Revised: November 9, 2025

Early access: January 16, 2026

Published online: January 16, 2026

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Abstract

Cyclic codes are widely employed in storage devices, communication systems, and consumer electronics. Their superiority lies in their clear algebraic structure, efficient coding, and decoding algorithms. BCH codes are a significant subclass of cyclic codes. We study a family of cyclic codes of length $ n $ over the finite field $ \mathbb{F}_q $ with $ n\mid (q+1) $. The dimensions of this family of cyclic codes gradually increase, whereas their minimum distances are kept constant at $ \frac{n}{2} $. This family of cyclic codes generates a family of non-narrow-sense BCH codes. The dual codes of non-narrow-sense BCH codes are both AMDS codes, distance-optimal and dimension-optimal locally repairable codes. Consequently, we determine the parameters of narrow-sense BCH codes with large designed distances and their dual codes. Meanwhile, we also give the parameters of the extended code of a classical family of linear complementary dual MDS codes.

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Mathematics Subject Classification: Primary: 94B05.

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Table 1. The dimensions $ k $ and minimum distances $ d $ of the cyclic code $ \mathcal{C} = \langle G_I \mathbb{M}_{\beta^4}(x)\cdots\mathbb{M}_{\beta^{11}}(x) \rangle $

$ I $	$ \emptyset $	{1}	{2}	{3}	{1, 2}	{1, 3}	{2, 3}	{1, 2, 3}
$ k $	8	6	6	6	4	4	4	2
$ d $	12	12	12	12	12	12	12	12

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Table 2. The first few largest coset leaders modulo $ q^m+1 $ for odd prime powers $ q $ and a positive integer $ m>1 $

coset leaders	expressions	the ranges of m	the sizes	the references
$ \delta_1 $	$ \frac{n}{2} $	any	1	[41]
$ \delta_2 $	$ \frac{(q-1)n}{2(q+1)} $	odd	2	[35]
	$ \Psi_q(l-1) $	$ m=2^l $ for $ l\geq1 $	$ 2m $	[28,39,41]
	$ \frac{n\Psi_q(l-1)}{q^{2^l}+1} $	$ m=2^l+h2^{l+1} $ for $ h, l\geq1 $	$ 2^{l+1} $	[28,39]
$ \delta_3 $	$ \frac{(q-1)(q^m-2q^{m-2}-1)}{2(q+1)} $	odd	$ 2m $	[35]
	$ \delta_2-2\Psi_q(l-3) $	$ m=2^l $ for $ l\geqslant 1 $	$ 2m $	[28,39,41]
	$ \delta_2-2\frac{\delta_2+\Psi_q(l)}{q^{2^l}+1} $	$ m=2^l+h2^{l+1} $ for $ h, l\geq1 $	$ 2m $	[28,39]
Note that when $ m $ is even, $ \delta_3 $ was not completely determined, which is still a conjecture. In this table, $ \begin{eqnarray} \Psi_q(x) & = & \left\{\begin{array}{ll} \frac{q-1}{2} \prod_{j=0}^{x}(q^{2^j}-1) , & \text { if } x\geqslant 0, \\ \frac{q-1}{2} , & \text { if } x=-1, \\ \frac{1}{2}, & \text { if } x=-2. \end{array}\right. \end{eqnarray} $

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