doi: 10.3934/amc.2020085

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 Published  June 2020

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

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} $.

Citation: Ranya Djihad Boulanouar, Aicha Batoul, Delphine Boucher. An overview on skew constacyclic codes and their subclass of LCD codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2020085
References:
[1]

A. BatoulK. Guenda and T. A. Gulliver, Some constacyclic codes over finite chain rings, Adv. Math. Commun., 10 (2016), 683-694.  doi: 10.3934/amc.2016034.  Google Scholar

[2]

D. BoucherW. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. Boucher, A first step towards the skew duadic codes, Adv. Math. Commun., 12 (2018), 3,553–577. doi: 10.3934/amc.2018033.  Google Scholar

[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.  Google Scholar

[8]

N. L. Fogarty, On Skew-Constacyclic Codes, Ph.D dissertation, University of Kentucky, 2016.  Google Scholar

[9]

M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symbolic Comput., 26 (1998), 463-486.  doi: 10.1006/jsco.1998.0224.  Google Scholar

[10]

C. GüneriaB. Özkayaa and P. Solé, Quasi-cyclic complementary dual codes, Finite Fields Appl., 42 (2016), 67-80.  doi: 10.1016/j.ffa.2016.07.005.  Google Scholar

[11]

C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., 86 (2018), 2261-2278.  doi: 10.1007/s10623-017-0447-0.  Google Scholar

[12]

C. LiC. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inform. Theory, 63 (2017), 4344-4356.  doi: 10.1109/TIT.2017.2672961.  Google Scholar

[13]

F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups, Ph.D. thesis, Radcliffe College, Cambridge, MA, 1962.  Google Scholar

[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.  Google Scholar

[15]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106-107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U.  Google Scholar

[16]

J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380.  doi: 10.1016/S0019-9958(64)90438-3.  Google Scholar

[17]

B. R. McDonald, Finite rings with identity, in Pure and Applied Mathematics, Vol. 28, Marcel Dekker Inc., New York, 1974.  Google Scholar

[18]

R. W. K. Odoni, On additive polynomials over a finite field, Proc. Edinburgh Math. Soc., 42 (1999), 1-16.  doi: 10.1017/S0013091500019970.  Google Scholar

[19]

O. Ore, Theory of non-commutative polynomials, Ann. Math., 34 (1933), 480-508.  doi: 10.2307/1968173.  Google Scholar

[20]

B. PangS. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. BatoulK. Guenda and T. A. Gulliver, Some constacyclic codes over finite chain rings, Adv. Math. Commun., 10 (2016), 683-694.  doi: 10.3934/amc.2016034.  Google Scholar

[2]

D. BoucherW. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. Boucher, A first step towards the skew duadic codes, Adv. Math. Commun., 12 (2018), 3,553–577. doi: 10.3934/amc.2018033.  Google Scholar

[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.  Google Scholar

[8]

N. L. Fogarty, On Skew-Constacyclic Codes, Ph.D dissertation, University of Kentucky, 2016.  Google Scholar

[9]

M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symbolic Comput., 26 (1998), 463-486.  doi: 10.1006/jsco.1998.0224.  Google Scholar

[10]

C. GüneriaB. Özkayaa and P. Solé, Quasi-cyclic complementary dual codes, Finite Fields Appl., 42 (2016), 67-80.  doi: 10.1016/j.ffa.2016.07.005.  Google Scholar

[11]

C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr., 86 (2018), 2261-2278.  doi: 10.1007/s10623-017-0447-0.  Google Scholar

[12]

C. LiC. Ding and S. Li, LCD cyclic codes over finite fields, IEEE Trans. Inform. Theory, 63 (2017), 4344-4356.  doi: 10.1109/TIT.2017.2672961.  Google Scholar

[13]

F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups, Ph.D. thesis, Radcliffe College, Cambridge, MA, 1962.  Google Scholar

[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.  Google Scholar

[15]

J. L. Massey, Linear codes with complementary duals, Discrete Math., 106-107 (1992), 337-342.  doi: 10.1016/0012-365X(92)90563-U.  Google Scholar

[16]

J. L. Massey, Reversible codes, Information and Control, 7 (1964), 369-380.  doi: 10.1016/S0019-9958(64)90438-3.  Google Scholar

[17]

B. R. McDonald, Finite rings with identity, in Pure and Applied Mathematics, Vol. 28, Marcel Dekker Inc., New York, 1974.  Google Scholar

[18]

R. W. K. Odoni, On additive polynomials over a finite field, Proc. Edinburgh Math. Soc., 42 (1999), 1-16.  doi: 10.1017/S0013091500019970.  Google Scholar

[19]

O. Ore, Theory of non-commutative polynomials, Ann. Math., 34 (1933), 480-508.  doi: 10.2307/1968173.  Google Scholar

[20]

B. PangS. 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.  Google Scholar

[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.  Google Scholar

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 $
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 $
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
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
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
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
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
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
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
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
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
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
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
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
[1]

Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39

[2]

Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273

[3]

Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039

[4]

Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034

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