Construction for both self-dual codes and LCD codes

Keita Ishizuka; Ken Saito

doi:10.3934/amc.2021070

Article Contents

2023, Volume 17, Issue 1: 139-151. Doi: 10.3934/amc.2021070

This issue Previous Article Automorphism groups and isometries for cyclic orbit codes Next Article The interplay of different metrics for the construction of constant dimension codes

Construction for both self-dual codes and LCD codes

Keita Ishizuka^, and
Ken Saito

Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan

^*Corresponding author: Keita Ishizuka
^*Corresponding author: Keita Ishizuka

Received: August 2021

Revised: November 2021

Early access: January 2022

Published: December 2022

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Abstract

From a given [n, k] code C, we give a method for constructing many [n, k] codes C^' such that the hull dimensions of C and C^' are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even [56, 28, 12] codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length 26 ≤ n ≤ 40.

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Mathematics Subject Classification: Primary: 94B05; Secondary: 94B65.

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Figure 1. Matrices $ A_{37,22,5},A_{38,13,10} $

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Table 1. Inequivalent extremal doubly even $ [56,28,12] $ codes

	y₄	y₈	y₁₂	y₁₆	y₂₀	y₂₄	total
D11	45	19	15	2	33	4	118
C_56,1	16	3	1	0	10	26	56
C_56,2	34	27	26	1	3	14	105
C_56,3	10	0	23	2	0	24	59
C_56,4	10	109	17	58	2	16	212
C_56,5	17	53	25	11	5	4	115

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Table 2. $ P_i,S_i $ for $ i = 1,2,3 $

i	P_i	S_i
1	{2,5}	{1,2,3,4,5,6,7,8,9,11}
2	{1,2,4}	{3}

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Table 3. $ C_{n,k,d} $ with $ x,y $

C_n,k,d	x	y
C_37,22,6	(010110011011111)	(110010110000001)
C_38,13,11	(0010011100110001011000100)	(1110100001110101110001101)

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Table 4. $ d_{LCD}(n,k) $, where $ 26 \le n \le 40, 9 \le k \le 17 $

n\k	9	10	11	12	13	14	15	16	17
26	9	8	8	8	7	6	5-6	5	4
27	9	9	8	8	7	6	6	6	5
28	10	10	8	8	8	7	6	6	5-6
29	10	10	9	8	8	8	6	6	6
30	11	10	9-10	9	8	8	6-7	6	6
31	11	10	10	10	9	8	7-8	6-7	6
32	12	11	10	10	9-10	8-9	7-8	7-8	6-7
33	12	12	10-11	10	10	9-10	8-9	8	6-8
34	13	12	11-12	10-12	10	10	9-10	8-9	7-8
35	13-14	12-13	12	10-12	10-11	10	9-10	9-10	7-8
36	14	12-14	12-13	11-12	10-12	10-11	10	10	8-9
37	14	12-14	12-14	12-13	10-12	10-12	10-11	10	9-10
38	14-15	13-14	12-14	12-14	11^*-12	10-12	10-12	10-11	9-10
39	14-16	14-15	13-14	12-14	11-13	11-12	10-12	10-12	10-11
40	15-16	14-16	13-15	13-14	12-14	11-13	10-12	10-12	10-12

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Table 5. d_LCD(n, k), where 26 ≤ n ≤ 40, 18 ≤ k ≤ 26

n\k	18	19	20	21	22	23	24	25	26
26	4	4	4
27	4	4	4	3
28	5	4	4	4	3
29	6	5	4	4	4	3
30	6	5	5	4	4	4	3
31	6	6	6	5	4	4	4	3
32	6	6	6	5-6	5	4	4	3-4	3
33	6-7	6	6	6	6	5	4	4	4
34	6-8	6-7	6	6	6	5-6	4	4	4
35	7-8	6-8	6-7	5-6	6	6	5	4	4
36	8	7-8	6-8	6-7	6	6	6	5	4
37	8-9	7-8	7-8	6-8	6^*-7	6	6	5-6	5
38	9-10	8-9	8	7-8	6-8	6-7	6	6	6
39	10	9-10	8-9	7-8	7-8	6-8	6-7	6	6
40	10-11	9-10	9-10	7-9	8	7-8	6-8	6-7	6

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Table 6. d_LCD(n, k), where 33 ≤ n ≤ 40, 27 ≤ k ≤ 34

n\k	27	28	29	30	31	32	33	34
33	3
34	3-4	3
35	4	4	3
36	4	4	3-4	3
37	4	4	4	4	3
38	5	4	4	4	3-4	3
39	5-6	5	4	4	4	4	2-3
40	6	6	5	4	4	4	3-4	2-3

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