Self-dual codes with an automorphism of order 13

Nikolay Yankov; Damyan Anev; Müberra Gürel

doi:10.3934/amc.2017047

Article Contents

2017, Volume 11, Issue 3: 635-645. Doi: 10.3934/amc.2017047

This issue Previous Article Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes Next Article

Self-dual codes with an automorphism of order 13

1.
Faculty of Mathematics and Informatics, Shumen University, 9700 Shumen, Bulgaria
2.
Istanbul Aydin University, Istanbul, Turkey

Received: February 2016

Revised: January 2017

Published: September 2017

This paper was supported by Shumen University under Grant RD-08-107/06.02.2017.

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Abstract

Using a method for constructing binary self-dual codes having an automorphism of odd prime order $p$ we classify, up to equivalence, all singly-even self-dual $[78,39,14]$, $[80,40,14]$, $[82,41,14],$ and $[84,42,14]$ codes as well as all doubly-even $[80,40,16]$ codes for $p=13$. The results show that there are exactly 1592 inequivalent binary self-dual $[78,39,14]$ codes with an automorphism of type $13-(6,0)$ and we found 6 new values of the parameter in the weight function thus increasing more than twice the number of known values. As for binary $[80,40]$ self-dual codes with an automorphism of type $13-(6,2)$ there are 162696 singly-even self-dual codes with minimum distance 14 and 195 doubly-even such codes with minimum distance 16. We also construct many new codes of lengths 82 and 84 with minimum distance 14. Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.

Keywords:

Mathematics Subject Classification: Primary: 11T71; Secondary: 94B05.

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Table 1. The number of codes with different $A_{16}$

$A_{16}$	#	$A_{16}$	#	$A_{16}$	#	$A_{16}$	#	$A_{16}$	#	$A_{16}$	#	$A_{16}$	#	$A_{16}$	#
14586	1	15002	359	15379	4280	15756	4993	16133	1458	16510	189	16887	16	17290	3
14612	1	15015	417	15392	4282	15769	4803	16146	1381	16523	144	16900	11	17303	4
14625	2	15028	528	15405	4380	15782	4902	16159	1198	16536	140	16913	11	17316	2
14651	1	15041	539	15418	4551	15795	4669	16172	1089	16549	144	16926	14	17342	1
14677	4	15054	608	15431	4766	15808	4414	16185	1111	16562	136	16939	11	17368	2
14690	1	15067	707	15444	4805	15821	4425	16198	1077	16575	113	16952	6	17394	2
14703	2	15080	735	15457	4842	15834	4215	16211	1007	16588	101	16965	12	17407	1
14716	3	15093	870	15470	5185	15847	4008	16224	856	16601	90	16978	7	17420	1
14729	9	15106	957	15483	5164	15860	3988	16237	876	16614	88	16991	7	17433	1
14742	4	15119	1031	15496	5366	15873	3865	16250	767	16627	70	17004	4	17446	3
14755	6	15132	1176	15509	5393	15886	3854	16263	762	16640	83	17017	3	17472	2
14768	8	15145	1277	15522	5480	15899	3611	16276	706	16653	59	17030	7	17485	2
14781	11	15158	1328	15535	5401	15912	3428	16289	662	16666	54	17043	8	17498	1
14794	18	15171	1557	15548	5568	15925	3369	16302	579	16679	59	17056	1	17511	1
14807	20	15184	1724	15561	5671	15938	3235	16315	541	16692	73	17069	4	17524	1
14820	37	15197	1809	15574	5617	15951	3027	16328	572	16705	50	17082	10	17537	1
14833	42	15210	1934	15587	5752	15964	2914	16341	508	16718	49	17095	3	17550	1
14846	30	15223	2092	15600	5577	15977	2751	16354	394	16731	41	17108	6	17563	2
14859	50	15236	2299	15613	5632	15990	2533	16367	438	16744	41	17121	4	17654	1
14872	53	15249	2439	15626	5669	16003	2531	16380	384	16757	38	17134	5	17693	2
14885	86	15262	2597	15639	5764	16016	2362	16393	312	16770	35	17147	2	17823	1
14898	99	15275	2848	15652	5552	16029	2328	16406	334	16783	33	17173	5	17888	1
14911	141	15288	2972	15665	5459	16042	2136	16419	279	16796	24	17186	3	18070	1
14924	152	15301	3066	15678	5353	16055	2075	16432	285	16809	19	17199	2	18083	1
14937	182	15314	3219	15691	5379	16068	1959	16445	243	16822	17	17212	2	18473	1
14950	199	15327	3487	15704	5482	16081	1812	16458	257	16835	23	17225	2	18655	1
14963	250	15340	3688	15717	5213	16094	1711	16471	241	16848	16	17251	3	18967	1
14976	271	15353	3875	15730	5359	16107	1597	16484	203	16861	13	17264	1	19071	1
14989	304	15366	4068	15743	5048	16120	1505	16497	208	16874	15	17277	3

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Table 2. The order of the automorphism groups of all $E_\sigma(C)^\ast$

$\|\text{Aut}(C)\|$	13	26	39	52	78	156	234	468
#	317529	4314	42	167	41	8	1	1

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Table 3. $\beta$ in $W_{78, 1}$ and $|\text{Aut}(C)|$ for $[78, 34, 14]$ self-dual codes with an automorphism of type $13-(6, 0)$

	$\|\text{Aut}(C)\|$					$\|\text{Aut}(C)\|$
$\beta$	13	26	39	78	$\beta$	13	26	39	78
$\boldsymbol{-117}$			$\textbf{1}$		$\boldsymbol{-39}$	$\textbf{302}$
$\boldsymbol{-104}$		$\textbf{1}$			$-26$	437	30
$-78$	5	7	1	3	$\boldsymbol{-13}$	$\textbf{421}$
$\boldsymbol{-65}$	$\textbf{37}$				0	171	18		5
$\boldsymbol{-52}$	$\textbf{137}$	$\textbf{14}$		$\textbf{2}$

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Table 4. The cardinality of the automorphism group for the doubly-even $[80, 40, 16]$ codes

$\|\text{Aut}(C)\|$	13	26	78	246480
#	172	18	4	1

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Table 5. Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$ for the singly-even $[80, 40, 14]$ self-dual codes

	$\|\text{Aut}(C)\|$					$\|\text{Aut}(C)\|$				$\|\text{Aut}(C)\|$
$\alpha$	13	26	39	78	$\alpha$	13	26	39	$\alpha$	13	26
$-351$		1	1		$-221$	6192	26		$-104$	7078	24
$-325$		1			$-208$	9815	45		$-91$	3632	3
$-312$	5	2		2	$-195$	14379	32	1	$-78$	1637	14
$-299$	32	1			$-182$	18952	44		$-65$	604	1
$-286$	121	4		2	$-169$	22002	31	1	$-52$	168	4
$-273$	294	7	1		$-156$	22611	42		$-39$	50
$-260$	702	11	3		$-143$	20758	13		$-26$	4
$-247$	1653	12			$-130$	16648	38
$-234$	3326	21	2		$-117$	11635	8

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Table 6. Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $\beta=0$ in $W_{82, 2}$ for the $[82, 41, 14]$ self-dual codes with an automorphism of type $13-(6, 4)$, $C_\pi=G_4$

	$\|\text{Aut}(C)\|$			$\|\text{Aut}(C)\|$				$\|\text{Aut}(C)\|$				$\|\text{Aut}(C)\|$
$\alpha$	13	26	$\alpha$	13	26	39	$\alpha$	13	26	39	$\alpha$	13	26	39
$-169$	1		$-260$	3199	79		$-351$	88751	312	1	$-442$	3002	28
$-182$		2	$-273$	7856	137		$-364$	82010	297	5	$-455$	1064	15
$-195$	4	2	$-286$	16765	168	1	$-377$	67138	223	1	$-468$	304	2
$-208$	17	8	$-299$	30265	230		$-390$	46912	174	2	$-481$	63	5	1
$-221$	88	13	$-312$	48818	302		$-403$	28921	143	2	$-494$	20
$-234$	411	19	$-325$	68292	330	2	$-416$	15253	88	1	$-507$	6	2
$-247$	1227	50	$-338$	83418	318	3	$-429$	7240	51	1	$-520$		1

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Table 7. Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $\beta=13$ in $W_{82, 2}$ for the $[82, 41, 14]$ self-dual codes with an automorphism of type $13-(6, 4)$, $C_\pi=G_4$

	$\|\text{Aut}(C)\|$		$\|\text{Aut}(C)\|$			$\|\text{Aut}(C)\|$			$\|\text{Aut}(C)\|$
$\alpha$	13	$\alpha$	13	39	$\alpha$	13	39	$\alpha$	13	39
-325	1	-377	44		-429	148		-481	38
-338	3	-390	71		-442	133		-494	21
-351	4	-403	123	1	-455	100		-507	4	1
-364	26	-416	146		-468	59	1	-520	3

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Table 8. Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $\beta=0$ in $W_{82, 2}$ for the $[82, 41, 14]$ self-dual codes with an automorphism of type $13-(6, 4)$, $C_\pi=G_5$

	$\|\text{Aut}(C)\|$				$\|\text{Aut}(C)\|$				$\|\text{Aut}(C)\|$					$\|\text{Aut}(C)\|$
$\alpha$	13	26	39	$\alpha$	13	26	39	$\alpha$	13	26	39	78	$\alpha$	13	26	39
$-209$	4			$-287$	3291	41		$-365$	23481	92			$-443$	26		1
$-222$	14			$-300$	6480	67		$-378$	20367	89			$-456$	447	14	1
$-235$	47	3		$-313$	11032	65	2	$-391$	15274	76			$-469$	156	3
$-248$	196	13		$-326$	16352	87		$-404$	9827	52			$-482$	36
$-261$	557	15		$-339$	20791	93		$-417$	5746	50		1	$-495$	9	1
$-274$	1446	34	1	$-352$	23588	117	2	$-430$	2837	19	2		$-508$	1

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Table 9. Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $\beta=13$ in $W_{82, 2}$ for the $[82, 41, 14]$ self-dual codes with an automorphism of type $13-(6, 4)$, $C_\pi=G_5$

	$\|\text{Aut}(C)\|$			$\|\text{Aut}(C)\|$			$\|\text{Aut}(C)\|$
$\alpha$	13	39	$\alpha$	13	26	$\alpha$	13	26	39
$-339$	1		$-417$	27		$-482$	12		5
$-365$	5	1	$-430$	20		$-495$	4
$-378$	6		$-443$	1218	23	$-508$	1
$-391$	7		$-456$	20		$-521$	2	1
$-404$	20	2	$-469$	17

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Table 10. Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $W_{84, 2}$ for the $[84, 42, 14]$ self-dual codes with an automorphism of type $13-(6, 6)$, $C_\pi=G_7$

	$\|\text{Aut}(C)\|$					$\|\text{Aut}(C)\|$					$\|\text{Aut}(C)\|$
$\alpha$	13	26	39	78	$\alpha$	13	26	39	78	$\alpha$	13	26	39
3038	2				3272	36692	171			3506	12054	126
3064	3				3298	56508	214	4		3532	5477	84	5
3090	32	1			3324	75114	280			3558	2154	51
3116	126	8			3350	87114	288			3584	742	46
3142	568	16	2		3376	87901	334	6		3610	202	12	1
3168	1602	31	1		3402	77847	314			3636	44	6
3194	4452	48			3428	59855	288			3662	13	4
3220	10558	70	3		3454	40586	214	5		3688	1	1
3246	21074	145		1	3480	24064	207		1

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Table 11. Number of codes with pairs $(\alpha, |\text{Aut}(C)|)$, $W_{84, 2}$ for the $[84, 42, 14]$ self-dual codes with an automorphism of type $13-(6, 6)$, $C_\pi=G_9$

	$\|\text{Aut}(C)\|$				$\|\text{Aut}(C)\|$				$\|\text{Aut}(C)\|$				$\|\text{Aut}(C)\|$
$\alpha$	13	26	39	$\alpha$	13	26	39	$\alpha$	13	26	39	$\alpha$	13	26
3040		1		3222	7240	51	1	3404	83418	318	3	3586	1227	50
3066	6	2		3248	15253	88	1	3430	68292	330	2	3612	411	19
3092	20			3274	28921	143	2	3456	48818	302		3638	88	13
3118	63	5	1	3300	46912	174	2	3482	30265	230		3664	17	8
3144	304	2		3326	67138	223	1	3508	16765	168	1	3690	4	2
3170	1064	15		3352	82010	297	5	3534	7856	137		3716		2
3196	3002	28		3378	88751	312	1	3560	3199	79		3742	1

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