Improved lower bounds for self-dual codes over $ \mathbb{F}_{11} $, $ \mathbb{F}_{13} $, $ \mathbb{F}_{17} $, $ \mathbb{F}_{19} $ and $ \mathbb{F}_{23} $

T. Aaron Gulliver; Masaaki Harada

doi:10.3934/amc.2022083

Article Contents

2024, Volume 18, Issue 5: 1259-1265. Doi: 10.3934/amc.2022083

This issue Previous Article Concrete analysis of approximate ideal-SIVP to decision ring-LWE reduction Next Article Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms

Improved lower bounds for self-dual codes over $ \mathbb{F}_{11} $, $ \mathbb{F}_{13} $, $ \mathbb{F}_{17} $, $ \mathbb{F}_{19} $ and $ \mathbb{F}_{23} $

T. Aaron Gulliver^1, and
Masaaki Harada^2, ,

1.
Department of Electrical and Computer Engineering, University of Victoria, , PO Box 1700, STN CSC, Victoria, BC V8W 2Y2, Canada
2.
Research Center for Pure and Applied Mathematics, , Graduate School of Information Sciences, Tohoku University, Sendai 980–8579, Japan

^*Corresponding author: Masaaki Harada
^*Corresponding author: Masaaki Harada

Received: June 2022

Revised: August 2022

Early access: October 12, 2022

Published online: October 12, 2022

The second author is supported by JSPS KAKENHI Grant Number 19H01802.

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Abstract

We construct self-dual codes over $ \mathbb{F}_{11} $, $ \mathbb{F}_{13} $, $ \mathbb{F}_{17} $, $ \mathbb{F}_{19} $ and $ \mathbb{F}_{23} $ which improve the previously known lower bounds on the largest minimum weights. In particular, the largest possible minimum weight among self-dual $ [n, n/2] $ codes over $ \mathbb{F}_{p} $ is determined for $ (p, n) = (19, 24) $ and $ (23, 28) $.

Keywords:

Mathematics Subject Classification: Primary: 94B05; Secondary: 94B25.

Citation:

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Table 1. New four-negacirculant self-dual codes $ C_{p, n, d} $

$ p $	$ n $	$ d $	$ r_A $	$ r_B $
11	24	10	$ (1, 3, 8, 0, 9, 7) $	$ (3, 5, 2, 8, 0, 1) $
11	28	11	$ (1, 5, 9, 4, 0, 4, 2) $	$ (1, 4, 10, 0, 0, 7, 3) $
13	28	12	$ (1, 10, 8, 7, 3, 9, 11) $	$ (2, 6, 10, 3, 7, 11, 10) $
13	32	13	$ (1, 3, 7, 3, 4, 12, 2, 8) $	$ (5, 11, 7, 2, 4, 3, 4, 3) $
13	36	14	$ (1, 11, 11, 10, 9, 5, 12, 3, 6) $	$ (9, 7, 11, 4, 9, 10, 12, 0, 2) $
13	40	15	$ (1, 4, 3, 6, 3, 12, 12, 9, 7, 11) $	$ (11, 0, 0, 8, 7, 4, 5, 8, 10, 4) $
17	24	11	$ (1, 5, 8, 11, 7, 7) $	$ (5, 12, 5, 12, 4, 4) $
17	28	12	$ (1, 10, 6, 13, 2, 2, 16) $	$ (16, 15, 1, 3, 14, 14, 5) $
17	32	13	$ (1, 6, 3, 7, 15, 16, 14, 5) $	$ (15, 13, 3, 12, 9, 6, 8, 15) $
17	36	14	$ (1, 10, 8, 10, 3, 2, 4, 1, 7) $	$ (0, 0, 8, 11, 3, 8, 3, 10, 11) $
17	40	15	$ (1, 5, 9, 2, 9, 16, 14, 14, 13, 16) $	$ (7, 0, 11, 0, 5, 12, 4, 12, 2, 13) $
19	24	12	$ (3, 9, 1, 0, 18, 10) $	$ (7, 10, 4, 3, 11, 5) $
19	28	12	$ (1, 5, 14, 10, 0, 12, 6) $	$ (15, 14, 1, 4, 9, 14, 6) $
23	28	14	$ (1, 11, 8, 15, 12, 22, 19) $	$ (22, 16, 15, 21, 15, 16, 22) $
23	32	13	$ (1, 3, 7, 7, 22, 3, 4, 9) $	$ (5, 2, 4, 22, 13, 7, 6, 17) $
23	36	15	$ (1, 17, 14, 9, 6, 5, 1, 20, 4) $	$ (8, 15, 19, 6, 18, 3, 20, 19, 16) $
23	40	16	$ (1, 17, 17, 17, 9, 4, 10, 15, 13, 1) $	$ (19, 20, 4, 16, 10, 22, 4, 21, 16, 21) $

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Table 2. New double circulant self-dual codes $ C_{p, n, d} $

$ p $	$ n $	$ d $	$ r_A $
13	26	11	$ (0, 0, 11, 9, 10, 11, 3, 9, 9, 7, 2, 1, 1) $
13	30	12	$ (12, 0, 7, 7, 5, 9, 12, 8, 11, 7, 4, 12, 0, 3, 12) $
13	38	14	$ (4, 0, 9, 9, 4, 2, 9, 3, 8, 9, 8, 9, 10, 0, 4, 8, 0, 2, 11) $

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Table 3. New quasi-twisted self-dual codes $ C_{p, n, d} $

$ p $	$ n $	$ d $	$ r_A $
13	34	13	$ (7, 12, 5, 1, 9, 2, 10, 7, 10, 12, 6, 12, 0, 0, 0, 3, 7) $
17	26	11	$ (1, 14, 9, 10, 8, 1, 12, 16, 0, 10, 2, 0, 15) $

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Table 4. Weight enumerators of self-dual $ [24, 12, 12] $ codes over $ \mathbb{F}_{19} $

$ i $	$ A_i $
0	$ 1 $
12	$ A_{12} $
13	$ 44930592 - 12 A_{12} $
14	$ 211815648 + 66 A_{12} $
15	$ 4377523392 - 220 A_{12} $
16	$ 39503618352 + 495 A_{12} $
17	$ 343122660144 - 792 A_{12} $
18	$ 2391276145104 + 924 A_{12} $
19	$ 13601985566400 - 792 A_{12} $
20	$ 61202848135968 + 495 A_{12} $
21	$ 209841090069600 - 220 A_{12} $
22	$ 515063659381920 + 66 A_{12} $
23	$ 806186749666752 - 12 A_{12} $
24	$ 604640049552288 + A_{12} $

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Table 5. Weight enumerators of self-dual $ [28, 14, 14] $ codes over $ \mathbb{F}_{23} $

$ i $	$ A_i $
0	1
14	$ A_{14} $
15	$ 823727520 - 14A_{14} $
16	$ 5354228880 + 91A_{14} $
17	$ 132753380760 - 364A_{14} $
18	$ 1623118737240 + 1001A_{14} $
19	$ 19155651715200 - 2002A_{14} $
20	$ 189055086620400 + 3003A_{14} $
21	$ 1585168436554560 - 3432A_{14} $
22	$ 11095536548416320 + 3003A_{14} $
23	$ 63679171473197280 - 2002A_{14} $
24	$ 291862645189315200 + 1001A_{14} $
25	$ 1027356593521514256 - 364A_{14} $
26	$ 2607905178181294992 + 91A_{14} $
27	$ 4249919552832161824 - 14A_{14} $
28	$ 3339222505567885376 + A_{14} $

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Table 6. $ d_p(n) $ $ (p\in\{11, 13, 17, 19, 23\}, n\in\{14, 16, \ldots, 40\}) $

$ n \backslash p $	11	13	17	19	23
14		8	7–8
16	8	8	8–9	8–9	9
18		9	10
20	10	10	10	11	10–11
22		10–11	10–11
24	10–12	10–12	11–12	12	13
26		11–13	11–13
28	11–14	12–14	12–14	12–14	14
30		12–15	12–15
32	12–16	13–16	13–16	14–16	13–16
34		13–17	13–17
36	13–18	14–18	14-18	14–18	15–18
38		14–19	14–19
40	14–20	15–20	15–20	15–20	16–20

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