Binary cyclic codes from three classes of sequences

Ying Zhang; Xianhong Xie; Tingting Wu; Wei Chen

doi:10.3934/amc.2026007

Article Contents

2026, Volume 22: 185-203. Doi: 10.3934/amc.2026007

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Binary cyclic codes from three classes of sequences

1.
School of Information and Computer, Anhui Agricultural University, 230036, Hefei, China
2.
School of Mathematical Sciences, Anhui University, 230601, Hefei, China

^*Corresponding author: Tingting Wu
^*Corresponding author: Tingting Wu

Received: March 20, 2025

Revised: October 17, 2025

Early access: December 17, 2025

Published online: December 17, 2025

This research is supported by [National Natural Science Foundation of China (Grant No.62402004, 62201009), and Natural Science Foundation of Anhui Province under Project 2508085QA033, and Natural Science Research Project of Anhui Educational Committee (Grant No. 2024AH050430), and in part by the Talent Research Foundation of Anhui Agricultural University, China (Grant No. rc482402) and Newly Established Quality Enhancement Project (Grant No. 2024xjzlts007)].

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Abstract

The objective of this paper is to present the construction of binary cyclic codes from three families of sequences and develop lower bounds on the minimum distances of these codes using the BCH bound. The results show that some of these codes have length $ n $, dimension $ (n\pm1)/2 $, minimum distance $ d $ satisfying $ d^2\geq n $, and the lower bounds on the minimum distances of certain codes are tight by extensive experiments.

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Mathematics Subject Classification: Primary: 94B15; Secondary: 94B05.

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Table 1. The known results on Code 3

$ f(x) $	Constraints	Code	Distance($ d\geq $)	Ref.
$ x^{2^t+3} $	$ m=2t+1\geq 7 $	$ \mathcal{C}_{s} $	8	[16, Th.8]
$ x^{2^h-1} $	$ m\equiv{1} \pmod{2}, h>2 $	$ \mathcal{C}_{s} $	$ 2^{h-2}+2 $	[16, Th.12]
$ x^{2^h-1} $	otherwise	$ \mathcal{C}_{s} $	$ 2^{h-2}+1 $	[16, Th.12]
$ x^{2^{\frac{m-1}{2}}+2^{\frac{m-1}{4}}-1} $	$ m\equiv{1} \pmod{8}\geq 9 $	$ \mathcal{C}_{s} $	$ 2^{\frac{m-1}{4}}+2 $	[16, Th.18]
$ x^{2^{\frac{m-1}{2}}+2^{\frac{m-1}{4}}-1} $	$ m\equiv{1} \pmod{5}\geq 9 $	$ \mathcal{C}_{s} $	$ 2^{\frac{m-1}{4}} $	[16, Th.18]
$ x^{2^{2^{2h}-2^h+1}} $	$ \gcd(m, h)=1 $	$ \mathcal{C}_{s} $	$ 2^h+2 $($ h\equiv{0} \pmod{2} $) or $ 2^h $ ($ h\equiv{1} \pmod{2} $)	[16, Th.23]
$ x+x^{r}+x^{2^h-1} $	$ m\equiv{1} \pmod{2}, h=0 $	$ \mathcal{C}_{s} $	8	[16, Th.26]
$ x+x^{r}+x^{2^h-1} $	$ m\equiv{0} \pmod{2}, h=0 $	$ \mathcal{C}_{s} $	3	[16, Th.26]
$ x^{2^m-2} $	$ m=2^le>2, 2\nmid e, l\geq 2 $	$ \mathcal{C}_{S, 1} $	$ 2^{\frac{m}{2}}+2 $	[17, Th.4]
	$ m=2^le>2, 2\nmid e, l=1 $	$ \mathcal{C}_{S, 1} $	$ 2^{\frac{m-2}{2}}+2 $
	$ m=2^le>2, 2\nmid e\geq 3 $	$ \mathcal{C}_{S, 0} $	$ 2^{\frac{m-2^l+2}{2}}+2 $
$ x+x^{2^{m}-2}+x^{2^{h}-1} $	$ m=2^{l}e, \; l\geq2, \;m\geq8, 0<h\leq\frac{m-4}{2}, \;2\mid h $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{\frac{m}{2}}+4 $	[17, Th.5]
	$ m=2^{l}e, \; l\geq2, \;m\geq8 $, $ 0<h\leq\frac{m-4}{2} $, $ 2\nmid h $ or $ m=4 $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{\frac{m}{2}}+2 $
	$ m=2^{l}e, \; l=1, \;m\geq10 $, $ 4\leq h\leq \frac{m-6}{2} $, $ 2\mid h $ or $ m=6 $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{\frac{m-2}{2}}+4 $
	$ m=2^{l}e, \; 2\nmid e, l=1, m\geq 10, 4<h\leq{\frac{m-6}{2}}, 2\nmid{h} $ or $ 1\leq h \leq 3 $ or $ m=6 $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{\frac{m-2}{2}}+2 $
	$ m=2^{l}e, \; 2\nmid e\geq 3, 0<h< 2^l $	$ \mathcal{C}_{D, 0} $	$ 2^{\frac{m-2^l+2}{2}}+2 $
	$ m=2^{l}e, \; 2\nmid e\geq 3, h=2^l $	$ \mathcal{C}_{D, 0} $	$ 2^{\frac{m-2^l+2}{2}} $
	$ m\equiv{1} \pmod{4}\geq5 $ $ 0<h\leq {\frac{m-3}{2}} $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{\frac{m-1}{2}}+4 $	[17, Th.6]
	$ m\equiv{1} \pmod{4}\geq5 $ $ 0<h\leq {\frac{m-3}{2}} $	$ \mathcal{C}_{\mathcal{D}, 0} $	$ 2^{\frac{m-1}{2}}+1 $ ($ 2\mid h $) or $ 2^{\frac{m-1}{2}}+3 $ ($ 2\nmid h $)	[17, Th.6]
	$ m\equiv{3} \pmod{4}\geq7 $, $ 0<h\leq {\frac{m-3}{2}} $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{\frac{m-1}{2}}+4 $($ 2\mid h $) or $ 2^{\frac{m-1}{2}}+2 $($ 2\nmid h $)	[17, Th.7]
	$ m\equiv{3} \pmod{4}\geq7 $, $ 0<h\leq {\frac{m-3}{2}} $	$ \mathcal{C}_{\mathcal{D}, 0} $	$ 2^{\frac{m-1}{2}}+1 $	[17, Th.7]
$ x^{2^m-2} $	$ m\equiv{1} \pmod{2} $	$ \mathcal{C}_{s} $	$ d^2-d+1\geq 2^m-1 $	[18, Th.4.2]
$ x^{2^{4i}+2^{3i}+2^{2i}+2^i-1} $	$ m=5i $	$ \mathcal{C}_{s} $	$ 2^i+1 (i \equiv{0} \pmod{2}) $ or $ 2^i+1 (i \equiv{0} \pmod{2}) $	[19, Th.6]
$ x^{2^{\frac{m}{2}}+2} $	$ m=2l $	$ \mathcal{C}_{s} $	$ 2(2\mid l) $ or $ 3 (2\nmid l) $	[20, Th.1]
$ x^{2^{\frac{m-1}{2}}+2^{\frac{3m-1}{4}}-1} $	$ m=4k+3 $	$ \mathcal{C}_{s} $	$ 2^{k+1}+2 $	[20, Th.5]
$ x^{2^h+1} $	$ m\equiv{1} \pmod{2} $, $ \gcd(m, h)=1 $	$ \mathcal{C}_{s} $	4	[21, Th.1]
$ x^{2^h+1} $	$ m\equiv{2} \pmod{4}, \gcd(m, h)=2 $	$ \mathcal{C}_{s} $	3	[21, Th.1]
$ x^{2^{2h}+2^h+1} $	$ m=4h, h\equiv{1} \pmod{2} $	$ \mathcal{C}_{s} $	3	[21, Th.6]

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Table 2. The binary cyclic codes constructed in this paper

$ f(x) $	Constraints	Code	Dimension	Distance($ d\geq $)	Ref.
$ x^{2^{t}+3} $	$ m=2t+1\geq5 $	$ \mathcal{C}_{\textbf{s}} $	$ 2^{m}-4m-2 $	$ 8 $	Th.3.1
	$ m=2t\geq8 $	$ \mathcal{C}_{\textbf{s}} $	$ 2^{m}-\frac{7m}{2}-2 $	$ 6 $
	$ m=4, 6 $	$ \mathcal{C}_{\textbf{s}} $	$ 2^{m}-\frac{7m}{2}-2 $	$ 8 $
$ x^{2^{m}-2} $	$ m=2^{l}e, \; l\geq2, \;m\geq8 $	$ \mathcal{C}_{\textbf{s}, 1} $	$ 2^{m-1}+m-2 $	$ 2^{\frac{m}{2}}+2 $	Th.4.7
	$ m=2^{l}e, \; l=1 $ or $ m=4 $	$ \mathcal{C}_{\textbf{s}, 1} $	$ 2^{m-1}+m-2 $	$ 2^{\frac{m-2}{2}}+2 $
	$ m=2^{l}e, \; e\geq3 $	$ \mathcal{C}_{\textbf{s}, 0} $	$ 2^{m-1} $	$ 2^{\frac{m-2^{l}+2}{2}}+2 $
	$ m\equiv{1} \pmod{4}\geq5 $	$ \mathcal{C}_{\textbf{s}, 1} $	$ 2^{m-1}+m-1 $	$ 2^{\frac{m-1}{2}}+2 $	Th.4.10
	$ m\equiv{1} \pmod{4}\geq5 $	$ \mathcal{C}_{\textbf{s}, 0} $	$ 2^{m-1} $	$ 2^{\frac{m-1}{2}}+3 $	Th.4.10
	$ m\equiv{3} \pmod{4}\geq7 $	$ \mathcal{C}_{\textbf{s}, 1} $	$ 2^{m-1}+m-1 $	$ 2^{\frac{m-1}{2}}+2 $	Th.4.11
	$ m\equiv{3} \pmod{4}\geq7 $	$ \mathcal{C}_{\textbf{s}, 0} $	$ 2^{m-1} $	$ 2^{\frac{m-1}{2}}+1 $	Th.4.11
$ x+x^{2^{m}-2}+x^{2^{h}-1} $	$ m=2^{l}e, \; l\geq2, \;m\geq8, 1<h\leq\frac{m-2}{2}, \;2\mid h\geq4 $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{m-1}+m-2 $	$ 2^{\frac{m}{2}}+4 $	Th.5.8
	$ m=2^{l}e, \; l\geq2, \;m\geq8, h=2 $ or $ 1<h\leq\frac{m-2}{2}, \;2\nmid h $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{m-1}+m-2 $	$ 2^{\frac{m}{2}}+2 $
	$ m=2^{l}e, \; l=1, \;m\geq14, 5\leq h\leq\frac{m-4}{2}, \;2\mid h $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{m-1}+m-2 $	$ 2^{\frac{m-2}{2}}+4 $
	$ m=2^{l}e, \; l=1, \;m\geq10, (1<h\leq4 $ or $ 5\leq h\leq\frac{m-4}{2}), \;2\nmid h $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{m-1}+m-2 $	$ 2^{\frac{m-2}{2}}+2 $
	$ m=2^{l}e, \; e\geq3, 1<h\leq2^{l} $	$ \mathcal{C}_{\mathcal{D}, 0} $	$ 2^{m-1} $	$ 2^{\frac{m-2^{l}+2}{2}}+2 $
	$ m\equiv{1} \pmod{4}\geq5 $, $ 1<h\leq {\frac{m-1}{2}} $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{m-1}+m-1 $	$ 2^{\frac{m-1}{2}}+2 $	Th.5.9
	$ m\equiv{1} \pmod{4}\geq5 $, $ 1<h\leq {\frac{m-1}{2}} $	$ \mathcal{C}_{\mathcal{D}, 0} $	$ 2^{m-1} $	$ 2^{\frac{m-1}{2}}+3 $ ($ 2\mid h $) or $ 2^{\frac{m-1}{2}}+1 $ ($ 2\nmid h \geq 3 $)	Th.5.9
	$ m\equiv{3} \pmod{4}\geq7 $, $ 1<h\leq {\frac{m-1}{2}} $	$ \mathcal{C}_{\mathcal{D}, 1} $	$ 2^{m-1}+m-1 $	$ 2^{\frac{m-1}{2}} $	Th.5.10
	$ m\equiv{3} \pmod{4}\geq7 $, $ 1<h\leq {\frac{m-1}{2}} $	$ \mathcal{C}_{\mathcal{D}, 0} $	$ 2^{m-1} $	$ 2^{\frac{m-1}{2}}+1 $ ($ 2\mid h $) or $ 2^{\frac{m-1}{2}}-1 $ ($ 2\nmid h \geq 3 $)	Th.5.10

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Table 3. The parameters of $ \mathcal{C}_{\textbf{s}, 1} $ and $ \mathcal{C}_{\textbf{s}, 0} $

$ m $	$ \dim(\mathcal{C}_{\textbf{s}, 1}) $	$ d(\mathcal{C}_{\textbf{s}, 1}) $	$ \dim(\mathcal{C}_{\textbf{s}, 0}) $	$ d(\mathcal{C}_{\textbf{s}, 0}) $
4	10	4	8	4
6	36	6	32	10
8	134	18	128	22
10	520	18	512	34
12	2058	66	2048	34
14	8204	66	8192	130
16	32782	258	32768	146

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Table 4. The minimum distances of $ \mathcal{C}_{\mathcal{D}, 1} $ and $ \mathcal{C}_{\mathcal{D}, 0} $

$ m $	$ \mathcal{C}_{\mathcal{D}, 1} $			$ \mathcal{C}_{\mathcal{D}, 0} $
$ m $	$ \dim(\mathcal{C}_{\mathcal{D}, 1}) $	$ h $	$ d(\mathcal{C}_{\mathcal{D}, 1}) $	$ \dim(\mathcal{C}_{\mathcal{D}, 0}) $	$ h $	$ d(\mathcal{C}_{\mathcal{D}, 0}) $
8	134	2, 3	18	128	2	22
8	134	2, 3	18	128	3	16
10	520	2, 3	18	512	2	34
10	520	4	28	512	2	34
12	2058	2, 3, 5	66	2048	2, 4	34
12	2058	4	68	2048	3	35
14	8204	2, 3, 5	66	8192	2	130
14	8204	4	76	8192	2	130
16	32782	1, 3, 5, 7 2, 4, 6	258 260	32768	2, 3, 5	146
					4, 6	142
					7	136

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