On the dimension of the subfield subcodes of 1-point Hermitian codes

Sabira El Khalfaoui; Gábor P. Nagy

doi:10.3934/amc.2020054

Article Contents

2021, Volume 15, Issue 2: 219-226. Doi: 10.3934/amc.2020054

This issue Previous Article Encryption scheme based on expanded Reed-Solomon codes Next Article Construction of minimal linear codes from multi-variable functions

On the dimension of the subfield subcodes of 1-point Hermitian codes

Sabira El Khalfaoui^1,2, and
Gábor P. Nagy^1,2,

1.
Bolyai Institute of the University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
2.
Department of Algebra of the Budapest University of Technology and Economics, Egry József utca 1, H-1111 Budapest, Hungary

Received: June 2019

Revised: September 2019

Early access: January 2020

Published: May 2021

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Abstract

Subfield subcodes of algebraic-geometric codes are good candidates for the use in post-quantum cryptosystems, provided their true parameters such as dimension and minimum distance can be determined. In this paper we present new values of the true dimension of subfield subcodes of $ 1 $–point Hermitian codes, including the case when the subfield is not binary.

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Mathematics Subject Classification: Primary: 11T71, 14G50; Secondary: 94B27.

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Table 1. Parameters of $ C_{8,2}(s) $ for $ s\in\{256,\ldots,511\} $

$ s $	$ \dim C_{8,2}(s) $	$ \dim \mathcal{H}(64,s) $	$ s $	$ \dim C_{8,2}(s) $	$ \dim \mathcal{H}(64,s) $
256	7	229	456	206	429
288	13	261	457	212	430
292	19	265	458	218	431
320	25	293	460	224	433
324	28	297	462	226	435
328	34	301	464	232	437
336	36	309	466	238	439
352	42	325	468	244	441
356	48	329	470	250	443
360	54	333	472	256	445
364	60	337	473	262	446
368	66	341	474	268	447
376	72	349	475	274	448
378	74	351	480	280	453
384	80	357	482	286	455
392	86	365	484	292	457
400	92	373	486	295	459
402	98	375	488	301	461
408	104	381	489	307	462
410	110	383	490	313	463
416	116	389	491	319	464
418	122	391	492	325	465
420	128	393	493	331	466
424	134	397	496	337	469
428	140	401	498	343	471
432	146	405	500	349	473
434	152	407	502	355	475
436	158	409	504	361	477
438	164	411	505	367	478
440	170	413	506	373	479
442	176	415	507	379	480
444	182	417	508	385	481
448	188	421	509	391	482
450	194	423	510	397	483
452	200	425	511	403	484

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