The error-correcting pair for direct sum codes

Boyi He; Qunying Liao

doi:10.3934/amc.2023046

Article Contents

2025, Volume 19, Issue 1: 180-201. Doi: 10.3934/amc.2023046

This issue Previous Article The MacWilliams identity for the skew rank metric Next Article Constructions of optimal Z-periodic complementary sequence sets with large zero correlation zone

The error-correcting pair for direct sum codes

Boyi He and
Qunying Liao^,

College of Mathematical Science, Sichuan Normal University, Chengdu Sichuan, 610066, China

^*Corresponding author: Qunying Liao
^*Corresponding author: Qunying Liao

Received: April 2023

Revised: October 2023

Early access: November 3, 2023

Published online: November 3, 2023

The second author is supported by [National Natural Science Foundation of China (12071321).].

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Abstract

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes. In this paper, we focus our study on the error-correcting pair for the direct sum code of two linear codes with an error-correcting pair. Firstly, for the direct sum code $ \mathcal{C} $ of two linear codes with an error-correcting pair, several sufficient conditions for $ \mathcal{C} $ with an error-correcting pair are given. Secondly, for the direct sum code $ \mathcal{C} $ of two Maximal Distance Separable linear codes, two Near-Maximal Distance Separable linear codes, or a Maximal Distance Separable linear code and a Near-Maximal Distance Separable linear code, several sufficient conditions for $ \mathcal{C} $ with an error-correcting pair are given, respectively. And then, we introduce the corresponding decoding procedure of the direct sum code with an error-correcting pair, and give several examples.

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Mathematics Subject Classification: Primary: 94B35; Secondary: 94B05.

Citation:

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Table 1. Conditions list for Corollary 5.4 (2)

Conditions	Parameters of $ \mathcal{A}_1 $	Parameters of $ \mathcal{A}_2 $	$ \sum\limits_{j=d_1}^{d_1+s_1}\!\!W_{1, j}+\!\!\!\!\sum\limits_{j=d_2}^{d_2+s_2}\!\!W_{2, j}\leq \!q^{k}\!-\!1 $
No.4.1	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $	(A.2)-(A.3)	$ s_1=s_2=0 $
No.4.2	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $	(A.4)-(A.6)	$ s_1=s_2=1 $

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Table 2. Conditions list for Corollary 5.5 (2)

Conditions	Parameters of $ \mathcal{A}_1 $	Parameters of $ \mathcal{A}_2 $	$ \sum\limits_{j = d_1}^{d_1+s_1}\!\!W_{1, j}+\!\!\!\!\sum\limits_{j = d_2}^{d_2+s_2}\!\!W_{2, j}\leq \!q^{k}\!-\!1 $
No.5.1	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $	(D.2)-(D.3)	$ s_1 = s_2 = 0 $
No.5.2	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $	(D.4)-(D.6)	$ s_1 = s_2 = 1 $
No.5.3	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $	(D.7)-(D.10)	$ s_1 = s_2 = 1 $

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Table 3. Conditions list for Corollary 5.6 (2)

Conditions	Parameters of $ \mathcal{A}_1 $	Parameters of $ \mathcal{A}_2 $	$ \sum\limits_{j = d_1}^{d_1+s_1}\!\!W_{1, j}+\!\!\!\!\sum\limits_{j = d_2}^{d_2+s_2}\!\!W_{2, j}\leq \!q^{k}\!-\!1 $
No.6.1	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $, $ [n_1, \ell_1\!+\!2, n_1\!-\!\ell_1\!-\!1] $	(D.2)-(D.3)	$ s_1 = s_2 = 0 $
No.6.2	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $, $ [n_1, \ell_1\!+\!2, n_1\!-\!\ell_1\!-\!1] $	(D.4)-(D.6)	$ s_1 = s_2 = 1 $
No.6.3	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1-1] $	(D.2)-(D.3)	$ s_1 = s_2 = 1 $
No.6.4	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1-1] $	(D.4)-(D.6)	$ s_1 = s_2 = 2 $

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Table 4. Conditions list for Corollary 5.7 (4)

Conditions	Parameters of $ \mathcal{A}_1 $	Parameters of $ \mathcal{A}_2 $	$ \sum\limits_{j = d_1}^{d_1+s_1}\!\!W_{1, j}+\!\!\!\!\sum\limits_{j = d_2}^{d_2+s_2}\!\!W_{2, j}\leq \!q^{k}\!-\!1 $
No.7.1	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $, $ [n_1, \ell_1\!+\!2, n_1\!-\!\ell_1\!-\!1] $	(D.4)-(D.6)	$ s_1 = s_2 = 0 $
No.7.2	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1] $, $ [n_1, \ell_1\!+\!2, n_1\!-\!\ell_1\!-\!1] $	(D.7)-(D.10)	$ s_1 = s_2 = 1 $
No.7.3	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1-1] $	(D.2)-(D.3)	$ s_1 = s_2 = 0 $
No.7.4	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1-1] $	(D.4)-(D.6)	$ s_1 = s_2 = 1 $
No.7.5	$ [n_1, \ell_1\!+\!1, n_1\!-\!\ell_1-1] $	(D.7)-(D.10)	$ s_1 = s_2 = 2 $

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Table 5. Conditions list for Corollary 5.8 (2)

Conditions	Parameters of $ \mathcal{A}_1 $	Parameters of $ \mathcal{A}_2 $	$ \sum\limits_{j=d_1}^{d_1+s_1}\!\!W_{1, j}+\!\!\!\!\sum\limits_{j=d_2}^{d_2+s_2}\!\!W_{2, j}\leq \!q^{k}\!-\!1 $
No.8.1	(A.1)	(A.2)-(A.3)	$ s_1=s_2=0 $
No.8.2	(A.1)	(A.4)-(A.6)	$ s_1=s_2=1 $
No.8.3	(A.2)-(A.3)	(A.1)	$ s_1=s_2=0 $
No.8.4	(A.2)-(A.3)	(A.2)-(A.3)	$ s_1=s_2=1 $
No.8.5	(A.2)-(A.3)	(A.4)-(A.6)	$ s_1=s_2=2 $
No.8.6	(A.4)-(A.6)	(A.1)	$ s_1=s_2=1 $
No.8.7	(A.4)-(A.6)	(A.2)-(A.3)	$ s_1=s_2=2 $
No.8.8	(A.4)-(A.6)	(A.4)-(A.6)	$ s_1=s_2=3 $

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Table 6. Conditions list for Corollary 5.9 (3)

Conditions	Parameters of $ \mathcal{A}_1 $	Parameters of $ \mathcal{A}_2 $	$ \sum\limits_{j=d_1}^{d_1+s_1}\!\!W_{1, j}+\!\!\!\!\sum\limits_{j=d_2}^{d_2+s_2}\!\!W_{2, j}\leq \!q^{k}\!-\!1 $
No.9.1	(D.1)	(D.4)-(D.6)	$ s_1=s_2=0 $
No.9.2	(D.1)	(D.7)-(D.10)	$ s_1=s_2=1 $
No.9.3	(D.2)-(D.3)	(D.2)-(D.3)	$ s_1=s_2=0 $
No.9.4	(D.2)-(D.3)	(D.4)-(D.6)	$ s_1=s_2=1 $
No.9.5	(D.2)-(D.3)	(D.7)-(D.10)	$ s_1=s_2=2 $
No.9.6	(D.4)-(D.6)	(D.1)	$ s_1=s_2=0 $
No.9.7	(D.4)-(D.6)	(D.2)-(D.3)	$ s_1=s_2=1 $
No.9.8	(D.4)-(D.6)	(D.4)-(D.6)	$ s_1=s_2=2 $
No.9.9	(D.4)-(D.6)	(D.7)-(D.10)	$ s_1=s_2=3 $
No.9.10	(D.7)-(D.10)	(D.1)	$ s_1=s_2=1 $
No.9.11	(D.7)-(D.10)	(D.2)-(D.3)	$ s_1=s_2=2 $
No.9.12	(D.7)-(D.10)	(D.4)-(D.6)	$ s_1=s_2=3 $
No.9.13	(D.7)-(D.10)	(D.7)-(D.10)	$ s_1=s_2=4 $

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Table 7. Conditions list for Corollary 5.10 (2)

Conditions	Parameters of $ \mathcal{A}_1 $	Parameters of $ \mathcal{A}_2 $	$ \sum\limits_{j=d_1}^{d_1+s_1}\!\!W_{1, j}+\!\!\!\!\sum\limits_{j=d_2}^{d_2+s_2}\!\!W_{2, j}\leq \!q^{k}\!-\!1 $
No.10.1	(A.1)	(D.2)-(D.3)	$ s_1=s_2=0 $
No.10.2	(A.1)	(D.4)-(D.6)	$ s_1=s_2=1 $
No.10.3	(A.1)	(D.7)-(D.10)	$ s_1=s_2=2 $
No.10.4	(A.2)-(A.3)	(D.1)	$ s_1=s_2=0 $
No.10.5	(A.2)-(A.3)	(D.2)-(D.3)	$ s_1=s_2=1 $
No.10.6	(A.2)-(A.3)	(D.4)-(D.6)	$ s_1=s_2=2 $
No.10.7	(A.2)-(A.3)	(D.7)-(D.10)	$ s_1=s_2=3 $
No.10.8	(A.4)-(A.6)	(D.1)	$ s_1=s_2=1 $
No.10.9	(A.4)-(A.6)	(D.2)-(D.3)	$ s_1=s_2=2 $
No.10.10	(A.4)-(A.6)	(D.4)-(D.6)	$ s_1=s_2=3 $
No.10.11	(A.4)-(A.6)	(D.7)-(D.10)	$ s_1=s_2=4 $

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