Self-dual and complementary dual abelian codes over Galois rings

Yıl 2019, Cilt: 6 Sayı: 2, 75 - 94, 07.05.2019

Somphong Jitman , San Ling

https://doi.org/10.13069/jacodesmath.560406

Öz

Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications.
In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${ GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${ GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${ GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $\gcd(|G|,p)=1$, the number of self-dual abelian codes in ${ GR}(p^r,s)[G]$ is completely and explicitly determined.
Applying known results on cyclic codes of length $p^a$ over ${ GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${ GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic.
Subsequently, the characterization and enumeration of complementary dual abelian codes in ${ GR}(p^r,s)[G]$ are established.
The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.

Anahtar Kelimeler

Abelian codes, Galois rings, Self-dual codes, Complementary dual codes, Codes over rings

Teşekkür

S. Jitman was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042. S. Ling was supported by Nanyang Technological University Research Grant M4080456.

Kaynakça

• [1] A. Batoul, K. Guenda, T. A. Gulliver, On self-dual cyclic codes over finite chain rings, Des. Codes Cryptogr. 70(3) (2014) 347–358.
• [2] S. Benson, Students ask the darnedest things: A result in elementary group theory, Math. Mag. 70(3) (1997) 207–211.
• [3] A. Boripan, S. Jitman, P. Udomkavanich, Characterization and enumeration of complementary dual abelian codes, J. Appl. Math. Comput. 58(1–2) (2018) 527–544.
• [4] A. Boripan, S. Jitman, P. Udomkavanich, Self-conjugate-reciprocal irreducible monic factors of $x^n-1$ over finite fields and their applications, Finite Fields Appl. 55 (2019) 78–96.
• [5] B. Chen, S. Ling, G. Zhang, Enumeration formulas for self-dual cyclic codes, Finite Fields Appl. 42 (2016) 1–22.
• [6] J. Chen, Y. Li, Y. Zhou, Morphic group rings, J. Pure Appl. Algebra 205(3) (2006) 621–639.
• [7] H.Q. Dinh, S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory 50(8) (2004) 1728–1744.
• [8] T. J. Dorsey, Morphic and principal-ideal group rings, J. Algebra 318(1) (2007) 393–411.
• [9] J. L. Fisher, S. K. Sehgal, Principal ideal group rings, Comm. Algebra 4(4) (1976) 319–325.
• [10] A. R. Hammons, P.V. Kumar, A. R. Calderbank, N. J. A. Sloane, P. Solé, The $\mathbb{Z}_4$ linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301–319.
• [11] Y. Jia, S. Ling, C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inform. Theory 57(4) (2011) 2243–2251.
• [12] S. Jitman, Good integers and some applications in coding theory, Cryptogr. Commun. 10(4) (2018) 685–704 and S. Jitman, Correction to: Good integers and some applications in coding theory, Cryptogr. Commun. 10(6) (2018) 1203–1203.
• [13] S. Jitman, S. Ling, H. Liu, X. Xie, Abelian codes in principal ideal group algebras, IEEE Trans. Inform. Theory 59(5) (2013) 3046–3058.
• [14] S. Jitman, S. Ling, E. Sangwisut, On self-dual cyclic codes of length $p^a$ over $\mathrm{ GR}({p^2},s)$, Adv. Math. Commun 10(2) (2016) 255–273.
• [15] S. Jitman, S. Ling, P. Solé, Hermitian self-dual abelian codes, IEEE Trans. Inform. Theory 60(3) (2014) 1496–1507.
• [16] H. M. Kiah, K. H. Leung, S. Ling, Cyclic codes over ${GR}(p^2,m)$ of length $p^k$, Finite Fields Appl. 14(3) (2008) 834–846.
• [17] H. M. Kiah, K. H. Leung, S. Ling, A note on cyclic codes over $GR(p^2,m)$ of length $p^k$, Des. CodesCryptogr. 63(1) (2012) 105–112.
• [18] T. Kiran, B. S. Rajan, Abelian codes over Galois rings closed under certain permutations, IEEE Trans. Inform. Theory 49(9) (2003) 2242–2253.
• [19] C. P. Milies, S. K. Sehgal, An Introduction to Group Rings, Lecture Notes in Mathematics vol. 1. Kluwer Academic Publishes, London, 2002.
• [20] P. Moree, On the divisors of $a^k+b^k$, Acta Arithm. 80 (1997) 197–212.
• [21] G. Nebe, E. M. Rains, N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics vol. 17, Springer-Verlag, Berlin 2006.
• [22] W. K. Nicholson, Local group rings, Canad. Math. Bull. 15(1) (1972) 137–138.
• [23] A. Salagean, Repeated-root cyclic and negacyclic codes over a finite chain ring, Discrete Appl. Math. 154(2) (2006) 413–419.
• [24] R. Sobhani, M. Esmaeili, A note on cyclic codes over ${GR}(p^2,m)$ of length $p^k$, Finite Fields Appl. 15(3) (2009) 387–391.
• [25] Z. X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific, New Jersey, 2003.
• [26] W. Willems, A note on self-dual group codes, IEEE Trans. Inform. Theory 48(12) (2002) 3107–3109.
• [27] X. Yang, J. L. Massey, The condition for a cyclic code to have a complementary dual, Discrete Math. 126(1–3) (1994) 391–393.