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Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices
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 |
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} $.
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. |
p | nbr of Euclidean LCD skew cyc. | nbr of Hermitian LCD skew cyc. | ||
3 | 18 | 16 | 36 | 32 |
5 | 3750 | 2412 | 3750 | 2412 |
7 | 705984 | 39564 | 941192 | 52752 |
11 | 259374246010 | 311249095212 |
p | nbr of Euclidean LCD skew cyc. | nbr of Hermitian LCD skew cyc. | ||
3 | 18 | 16 | 36 | 32 |
5 | 3750 | 2412 | 3750 | 2412 |
7 | 705984 | 39564 | 941192 | 52752 |
11 | 259374246010 | 311249095212 |
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 |
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 |
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 |
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 |
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 |
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 |
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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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