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November  2021, 15(4): 611-632. doi: 10.3934/amc.2020085

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

Ranya Djihad Boulanouar 1, , Aicha Batoul 1, and  Delphine Boucher 2,

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  November 2021 Early access  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} $.

Keywords: Skew polynomial rings, skew constacyclic codes, LCD codes, MDS codes, error-correcting codes, finite fields.
Mathematics Subject Classification: 94B05, 12-08, 68W30, 12E15.
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, 2021, 15 (4) : 611-632. doi: 10.3934/amc.2020085
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.

show all references

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.

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 $
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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
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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
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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
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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
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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
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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
Table Options

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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
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[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
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