An overview on skew constacyclic codes and their subclass of LCD codes

Ranya Djihad Boulanouar; Aicha Batoul; Delphine Boucher

doi:10.3934/amc.2020085

Article Contents

2021, Volume 15, Issue 4: 611-632. Doi: 10.3934/amc.2020085

This issue Previous Article Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices Next Article Quasi-symmetric designs on

$ 56 $

points

An overview on skew constacyclic codes and their subclass of LCD codes

1.
Faculty of Mathematics, University of Science and Technology Houari Boumedienne (USTHB), 16111 Bab Ezzouar, Algiers, Algeria
2.
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Received: September 2019

Revised: February 2020

Early access: June 2020

Published: November 2021

The third author is supported by the French government Investissements d'Avenir program ANR-11-LABX-0020-01

Abstract / Introduction Full Text(HTML) Figure(0) / Table(7) Related Papers Cited by

Abstract

This paper is about a first characterization of LCD skew constacyclic codes and some constructions of LCD skew cyclic and skew negacyclic codes over $ \mathbb{F}_{p^2} $.

Keywords:

Mathematics Subject Classification: 94B05, 12-08, 68W30, 12E15.

Citation:

Full Text(HTML)

Table 1. Number of Euclidean and Hermitian LCD $ [2p,p]_{p^2} $ and $ [2p,p]_{p^2} $ MDS skew-cyclic codes for $ p = 3,5,7,11 $

p	nbr of Euclidean LCD skew cyc.		nbr of Hermitian LCD skew cyc.
	$ [2p,p]_{p^2} $	$ [2p,p,p+1]_{p^2} $	$ [2p,p]_{p^2} $	$ [2p,p,p+1]_{p^2} $
3	18	16	36	32
5	3750	2412	3750	2412
7	705984	39564	941192	52752
11	259374246010	$ \geq 1 $	311249095212	$ \geq 1 $

| Show Table

DownLoad: CSV

Table 5. Dimensions of MDS LCD skew codes over $ {\mathbb F}_9 $ with length $ n \leq 10 $ and dimension $ 1 < k < n-1 $

	MDS Euclidean LCD		MDS Hermitian LCD
length	skew cyc	skew nega	skew cyc	skew nega
4	2	2	no	2
6	3	3	3	no
8	3, 4, 5	4	3, 5	4
10	5	5	5	no

| Show Table

DownLoad: CSV

Table 6. Dimensions of MDS LCD skew codes over $ {\mathbb F}_{25} $ with length $ n \leq 18 $ and dimension $ 1 < k < n-1 $

	MDS Euclidean LCD		MDS Hermitian LCD
length	skew cyc	skew nega	skew cyc	skew nega
4	2	2	no	2
6	2, 3, 4	2, 3, 4	2, 3, 4	2, 4
8	3, 4, 5	4	3, 5	4
10	5	5	5	no
12	3, 5, 6, 7, 9	6	3, 5, 7, 9	6
14	7	7	7	no
16	7, 8, 9	no	7, 9	no
18	9	9	9	no

| Show Table

DownLoad: CSV

Table 7. Dimensions of MDS LCD skew codes over $ {\mathbb F}_{49} $ with length $ n \leq 16 $ and dimension $ 1 < k < n-1 $

	MDS Euclidean LCD		MDS Hermitian LCD
length	skew cyc	skew nega	skew cyc	skew nega
4	2	2	no	2
6	2, 3, 4	2, 3, 4	2, 3, 4	2, 4
8	3, 4, 5	2, 4, 6	3, 5	2, 4, 6
10	4, 5, 6	4, 5, 6	4, 5, 6	4, 6
12	3, 5, 6, 7, 9	6	3, 5, 7, 9	6
14	7	7	7	no
16	3, 5, 7, 8, 9, 11, 13	8	3, 5, 7, 9, 11, 13	8

| Show Table

DownLoad: CSV

Table 2. Best minimum distances and numbers of Euclidean LCD $ [2k,k] $ skew cyclic codes of length $ \leq 48 $ over $ {\mathbb F}_4 $ with skew generator polynomial not divisible by a central polynomial

Euclidean LCD skew cyc.			Euclidean LCD skew cyc.
length	best dist	nbr	length	best dist	nbr
2	2^*	2	26	9	8 064
4	3^*	4	28	11^*	18 432
6	4^*	4	30	12^*	13 056
8	4^*	16	32	10	65 536
10	5^*	24	34	11^*	115 200
12	5	32	36	11	114 688
14	6^*	144	38	12^*	523 264
16	6	256	40	12^*	786 432
18	7	224	42	12	1 198 080
20	8^*	768	44	13	4 063 232
22	8^*	1 984	46	14^*	8 392 704
24	9^*	2 048	48	14^*	8 388 608

| Show Table

DownLoad: CSV

Table 3. Best minimum distances and numbers of LCD $ [2k,k] $ skew codes of length $ \leq 24 $ over $ {\mathbb F}_9 $ with skew generator polynomial not divisible by a central polynomial

	Euclidean LCD
	skew cyc		skew negacyc
length	best dist	nbr	best dist	nbr
2	2^*	2	2^*	4
4	3^*	32	3^*	6
6	4^*	18	4^*	36
8	5^*	192	5^*	90
10	6^*	144	6^*	288
12	6^*	5 408	6^*	486
14	7^*	1 404	7^*	2 808
16	7	17 280	8^*	6 642
18	9^*	13 122	9^*	26 244
20	9	165 888	9	39 852
22	9^*	118 584	9^*	237 168
24	10^*	2 628 288	10^*	590 490

| Show Table

DownLoad: CSV

Table 4. Best minimum distances and numbers of LCD $ [2k,k] $ skew codes of length $ \leq 24 $ over $ {\mathbb F}_9 $ with skew generator polynomial not divisible by a central polynomial

	Hermitian LCD
	skew cyc		skew negacyc
length	best dist	nbr	best dist	nbr
2	2^*	4	0	0
4	0	0	3^*	6
6	4^*	361	0	0
8	0	0	5^*	90
10	6^*	288	0	0
12	0	0	6^*	486
14	7^*	2 808	0	0
16	0	0	8^*	6 642
18	9^*	26 244	0	0
20	0	0	10^*	39 852
22	9^*	237 168	0	0
24	0	0	10^*	590 490

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	A. Batoul, K. Guenda and T. A. Gulliver, Some constacyclic codes over finite chain rings, Adv. Math. Commun., 10 (2016), 683-694. doi: 10.3934/amc.2016034.
[2]	D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.
[3]	D. Boucher and F. Ulmer, A note on the dual codes of module skew codes, in Cryptography and Coding, Vol. 7089, Lecture Notes in Comput. Sci., 2011,230–243. doi: 10.1007/978-3-642-25516-8_14.
[4]	D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials, J. Symbolic. Comput., 60 (2014), 47-61. doi: 10.1016/j.jsc.2013.10.003.
[5]	D. Boucher, Construction and number of self-dual skew codes over $ \mathbb{F}_{p^{2}} $, Adv. Math. Commun., 10 (2016), 4,765–795. doi: 10.3934/amc.2016040.
[6]	D. Boucher, A first step towards the skew duadic codes, Adv. Math. Commun., 12 (2018), 3,553–577. doi: 10.3934/amc.2018033.
[7]	C. Carlet, S. Mesnager, C. Tang and Y. Qi, Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr. 86 (2018), 11, 2605–2618. doi: 10.1007/s10623-018-0463-8.
[8]	N. L. Fogarty, On Skew-Constacyclic Codes, Ph.D dissertation, University of Kentucky, 2016.
[9]	M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symbolic Comput., 26 (1998), 463-486. doi: 10.1006/jsco.1998.0224.
[10]	C. Güneria, B. Özkayaa and P. Solé, Quasi-cyclic complementary dual codes, Finite Fields Appl., 42 (2016), 67-80. doi: 10.1016/j.ffa.2016.07.005.
[11]	C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., 86 (2018), 2261-2278. doi: 10.1007/s10623-017-0447-0.
[12]	C. Li, C. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inform. Theory, 63 (2017), 4344-4356. doi: 10.1109/TIT.2017.2672961.
[13]	F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups, Ph.D. thesis, Radcliffe College, Cambridge, MA, 1962.
[14]	J. L. Massey and X. Yang, The condition for a cyclic code to have a complementary dual, Discrete Math., 126 (1994), 391-393. doi: 10.1016/0012-365X(94)90283-6.
[15]	J. L. Massey, Linear codes with complementary duals, Discrete Math., 106-107 (1992), 337-342. doi: 10.1016/0012-365X(92)90563-U.
[16]	J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380. doi: 10.1016/S0019-9958(64)90438-3.
[17]	B. R. McDonald, Finite rings with identity, in Pure and Applied Mathematics, Vol. 28, Marcel Dekker Inc., New York, 1974.
[18]	R. W. K. Odoni, On additive polynomials over a finite field, Proc. Edinburgh Math. Soc., 42 (1999), 1-16. doi: 10.1017/S0013091500019970.
[19]	O. Ore, Theory of non-commutative polynomials, Ann. Math., 34 (1933), 480-508. doi: 10.2307/1968173.
[20]	B. Pang, S. Zhu and J. Li, On LCD repeated-root cyclic codes over finite fields, J. Appl. Math. Comput., 56 (2018), 625-635. doi: 10.1007/s12190-017-1118-z.
[21]	A. Sharma and T. Kaur, Enumeration formulae for self-dual, self-orthogonal and complementary-dual quasi-cyclic codes over finite fields, Cryptogr. Commun., 10 (2018), 401-435. doi: 10.1007/s12095-017-0228-7.