May  2019, 13(2): 281-296. doi: 10.3934/amc.2019019

Some new constructions of isodual and LCD codes over finite fields

1. 

University M'Hamed Bougara of Boumerdes, Faculty of Sciences, Boumerdes, Algeria

2. 

University of Science and Technology: Houari Boumediene, Faculty of Mathematics, Algiers, 16411, Algeria

3. 

Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC V8W 2Y2, Canada

* Corresponding author: Aicha Batoul

Received  April 2018 Revised  August 2018 Published  February 2019

This paper presents some new constructions of LCD, isodual, self-dual and LCD-isodual codes based on the structure of repeated-root constacyclic codes. We first characterize repeated-root constacyclic codes in terms of their generator polynomials and lengths. Then we provide simple conditions on the existence of repeated-root codes which are either self-dual negacyclic or LCD cyclic and negacyclic. This leads to the construction of LCD, self-dual, isodual, and LCD-isodual cyclic and negacyclic codes.

Citation: Fatma-Zohra Benahmed, Kenza Guenda, Aicha Batoul, Thomas Aaron Gulliver. Some new constructions of isodual and LCD codes over finite fields. Advances in Mathematics of Communications, 2019, 13 (2) : 281-296. doi: 10.3934/amc.2019019
References:
[1]

C. BachocT. A. Gulliver and M. Harada, Isodual codes over $Z_2k$ and isodual lattices, J. Algebra. Combin., 12 (2000), 223-240.  doi: 10.1023/A:1011259823212.

[2]

G. K. Bakshi and M. Raka, A class of constacyclic codes over a finite field, Finite Fields Appl., 18 (2012), 362-377.  doi: 10.1016/j.ffa.2011.09.005.

[3]

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.

[4]

A. Batoul, K. Guenda and T. A. Gulliver, Repeated-root isodual cyclic codes over finite fields, in Codes, Cryptology and Information Security (eds. S. El Hajji et al.), Lecture Notes in Computer Science, 9084, Springer, Cham, (2015), 119–132. doi: 10.1007%2F978-3-319-18681-8_10.

[5]

A. Batoul, K. Guenda, T. A. Gulliver and N. Aydin, On isodual cyclic codes over finite chain rings, in Codes, Cryptology and Information Security (eds. S. El Hajji et al.), Lecture Notes in Computer Science, 10194, Springer, Cham, (2017), 176–194. doi: 10.1007/978-3-319-55589-8_12.

[6]

S. D. Berman, Semisimple cyclic and abelian codes Ⅱ, Cybernetics and Systems Analysis, 3 (1967), 17-23. 

[7]

T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943.  doi: 10.1016/j.ffa.2008.05.004.

[8]

C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun., 10 (2016), 131-150.  doi: 10.3934/amc.2016.10.131.

[9]

C. CarletS. MesnagerC. TangY. Qi and R. Pellikaan, Linear codes over $ \mathbb{F}_q$ equivalent to LCD codes for $q>3$, IEEE Trans. Inform. Theory, 64 (2018), 3010-3017.  doi: 10.1109/TIT.2018.2789347.

[10]

B. ChenY. FanL. Lin and H. Liu, Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231.  doi: 10.1016/j.ffa.2012.10.001.

[11]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.

[12]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3{p^s}$ and their duals, Discrete Math., 313 (2013), 983-991.  doi: 10.1016/j.disc.2013.01.024.

[13]

H. Q. Dinh, Repeated-root cyclic and negacyclic codes of length $6{p^s}$, Contemporary Mathematics, 609 (2014), 69-87.  doi: 10.1090/conm/609.

[14]

G. FalknerB. KowolW. Heise and E. Zehendner, On the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979), 326-341. 

[15]

M. Grassl, Code tables: Bounds on the parameters of various types of codes, Available from: http://www.codetables.de.

[16]

K. Guenda, Dimension and minimum distance of a class of BCH codes, Annales Des Sciences Mathmatiques du Québec, 32 (2008), 57–62. Available from: http://www.labmath.uqam.ca/~annales/volumes/32-1/PDF/057-062.pdf.

[17]

K. Guenda, New MDS self-dual codes over finite fields, Des., Codes and Crypt., 62 (2012), 31-42.  doi: 10.1007/s10623-011-9489-x.

[18]

K. Guenda and T. A. Gulliver, Self-dual repeated-root cyclic and negacyclic codes over finite fields, Proc. IEEE Int. Symp. Inform. Theory, (2012), 2904–2908.

[19]

K. Guenda and T. A. Gulliver, Repeated-root constacyclic codes of length $mp^s$ over $ \mathbb{F}_{p^r}+u \mathbb{F}_{p^r}+ u^{e-1} \mathbb{F}_{p^r}$, Journal of Algebra and Its Applications, 14 (2015), 1450081, 1–12.

[20] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, New York, USA, 2003.  doi: 10.1017/CBO9780511807077.
[21]

S. LiC. LiC. Ding and H. Liu, Two families of LCD BCH codes, IEEE Trans. Inform. Theory, 63 (2017), 5699-5717.  doi: 10.1109/TIT.2017.2723363.

[22]

X. LiuY. Fan and H. Liu, Galois LCD codes over finite fields, Finite Fields and Applications, 49 (2018), 227-242.  doi: 10.1016/j.ffa.2017.10.001.

[23]

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

[24]

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.

[25]

J. H. Van Lint, Repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 343-345.  doi: 10.1109/18.75250.

[26]

X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discr. Math., 126 (1994), 391-393.  doi: 10.1016/0012-365X(94)90283-6.

show all references

References:
[1]

C. BachocT. A. Gulliver and M. Harada, Isodual codes over $Z_2k$ and isodual lattices, J. Algebra. Combin., 12 (2000), 223-240.  doi: 10.1023/A:1011259823212.

[2]

G. K. Bakshi and M. Raka, A class of constacyclic codes over a finite field, Finite Fields Appl., 18 (2012), 362-377.  doi: 10.1016/j.ffa.2011.09.005.

[3]

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.

[4]

A. Batoul, K. Guenda and T. A. Gulliver, Repeated-root isodual cyclic codes over finite fields, in Codes, Cryptology and Information Security (eds. S. El Hajji et al.), Lecture Notes in Computer Science, 9084, Springer, Cham, (2015), 119–132. doi: 10.1007%2F978-3-319-18681-8_10.

[5]

A. Batoul, K. Guenda, T. A. Gulliver and N. Aydin, On isodual cyclic codes over finite chain rings, in Codes, Cryptology and Information Security (eds. S. El Hajji et al.), Lecture Notes in Computer Science, 10194, Springer, Cham, (2017), 176–194. doi: 10.1007/978-3-319-55589-8_12.

[6]

S. D. Berman, Semisimple cyclic and abelian codes Ⅱ, Cybernetics and Systems Analysis, 3 (1967), 17-23. 

[7]

T. Blackford, Negacyclic duadic codes, Finite Fields Appl., 14 (2008), 930-943.  doi: 10.1016/j.ffa.2008.05.004.

[8]

C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun., 10 (2016), 131-150.  doi: 10.3934/amc.2016.10.131.

[9]

C. CarletS. MesnagerC. TangY. Qi and R. Pellikaan, Linear codes over $ \mathbb{F}_q$ equivalent to LCD codes for $q>3$, IEEE Trans. Inform. Theory, 64 (2018), 3010-3017.  doi: 10.1109/TIT.2018.2789347.

[10]

B. ChenY. FanL. Lin and H. Liu, Constacyclic codes over finite fields, Finite Fields Appl., 18 (2012), 1217-1231.  doi: 10.1016/j.ffa.2012.10.001.

[11]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.

[12]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3{p^s}$ and their duals, Discrete Math., 313 (2013), 983-991.  doi: 10.1016/j.disc.2013.01.024.

[13]

H. Q. Dinh, Repeated-root cyclic and negacyclic codes of length $6{p^s}$, Contemporary Mathematics, 609 (2014), 69-87.  doi: 10.1090/conm/609.

[14]

G. FalknerB. KowolW. Heise and E. Zehendner, On the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979), 326-341. 

[15]

M. Grassl, Code tables: Bounds on the parameters of various types of codes, Available from: http://www.codetables.de.

[16]

K. Guenda, Dimension and minimum distance of a class of BCH codes, Annales Des Sciences Mathmatiques du Québec, 32 (2008), 57–62. Available from: http://www.labmath.uqam.ca/~annales/volumes/32-1/PDF/057-062.pdf.

[17]

K. Guenda, New MDS self-dual codes over finite fields, Des., Codes and Crypt., 62 (2012), 31-42.  doi: 10.1007/s10623-011-9489-x.

[18]

K. Guenda and T. A. Gulliver, Self-dual repeated-root cyclic and negacyclic codes over finite fields, Proc. IEEE Int. Symp. Inform. Theory, (2012), 2904–2908.

[19]

K. Guenda and T. A. Gulliver, Repeated-root constacyclic codes of length $mp^s$ over $ \mathbb{F}_{p^r}+u \mathbb{F}_{p^r}+ u^{e-1} \mathbb{F}_{p^r}$, Journal of Algebra and Its Applications, 14 (2015), 1450081, 1–12.

[20] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, New York, USA, 2003.  doi: 10.1017/CBO9780511807077.
[21]

S. LiC. LiC. Ding and H. Liu, Two families of LCD BCH codes, IEEE Trans. Inform. Theory, 63 (2017), 5699-5717.  doi: 10.1109/TIT.2017.2723363.

[22]

X. LiuY. Fan and H. Liu, Galois LCD codes over finite fields, Finite Fields and Applications, 49 (2018), 227-242.  doi: 10.1016/j.ffa.2017.10.001.

[23]

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

[24]

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.

[25]

J. H. Van Lint, Repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 343-345.  doi: 10.1109/18.75250.

[26]

X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discr. Math., 126 (1994), 391-393.  doi: 10.1016/0012-365X(94)90283-6.

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