# Self-dual codes over F_q+uF_q+u^2 F_q and applications

### Abstract

Self-dual codes over finite fields and over some finite rings have been of interest and extensively studied due to their nice algebraic structures and wide applications. Recently, characterization and enumeration of Euclidean self-dual linear codes over the ring~$\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ with $u^3=0$ have been established. In this paper, Hermitian self-dual linear codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}$ are studied for all square prime powers~$q$. Complete characterization and enumeration of such codes are given. Subsequently, algebraic characterization of $H$-quasi-abelian codes in $\mathbb{F}_q[G]$ is studied, where $H\leq G$ are finite abelian groups and $\mathbb{F}_q[H]$ is a principal ideal group algebra. General characterization and enumeration of $H$-quasi-abelian codes and self-dual $H$-quasi-abelian codes in $\mathbb{F}_q[G]$ are given. For the special case where the field characteristic is $3$, an explicit formula for the number of self-dual $A\times \mathbb{Z}_3$-quasi-abelian codes in $\mathbb{F}_{3^m}[A\times \mathbb{Z}_3\times B]$ is determined for all finite abelian groups $A$ and $B$ such that $3\nmid |A|$ as well as their construction. Precisely, such codes can be represented in terms of linear codes and self-dual linear codes over $\mathbb{F}_{3^m}+u\mathbb{F}_{3^m}+u^2\mathbb{F}_{3^m}$. Some illustrative examples are provided as well.

### References

S. Benson, Students ask the darnedest things: A result in elementary group theory, Math. Mag. 70 (1997) 207–211.

S. D. Berman, Semi-simple cyclic and abelian codes, Kibernetika 3 (1967) 21–30.

R. A. Betty, F. Nemenzo, T. L. Vasques, Mass formula for self-dual codes over $mathbb{F}_{q}+umathbb{F}_{q}+u^2mathbb{F}_{q}$, J. Appl. Math. Comput. 57 (2018) 523–546.

Y. Cao, Y. Gao, Repeated root cyclic $mathbb{F}_q$-linear codes over $mathbb{F}_{q^l}$, Finite Fields Appl. 31 (2015) 202–227. [5] C. Ding, D. R. Kohel, S. Ling, Split group codes, IEEE Trans. Inform. Theory 46 (2000) 485–495.

J. L. Fisher, S. K. Sehgal, Principal ideal group rings, Comm. Algebra 4 (1976) 319–325.

A. R. Hammons Jr., 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 (1994) 301–319.

Y. Jia, S. Ling, C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inform. Theory 57 (2011) 2243–2251.

S. Jitman, Good integers and some applications in coding theory, Cryptogr. Commun. 10 (2018) 685–704. S. Jitman, Correction to: Good integers and some applications in coding theory, Cryptogr. Commun. 10 (2018) 1203–1203.

S. Jitman, S. Ling, Quasi-abelian codes, Des. Codes Cryptogr. 74 (2015) 511–531.

S. Jitman, S. Ling, H. Liu, X. Xie, Abelian codes in principal ideal group algebras, IEEE Trans. Info. Theory 59 (2013 ) 3046–3058.

S. Jitman, S. Ling, P. Solé, Hermitian self-dual abelian codes, IEEE Trans. Info. Theory 60 (2014) 1496–1507.

J.-L. Kim, Y. Lee, Euclidean and Hermitian self-dual MDS codes over large finite fields, J. Combinat. Theory A 105 (2004) 79–95.

F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, NorthHolland, Amsterdam, 1977.

C. P. Milies, S. K. Sehgal, S. Sehgal, An Introduction to Group Rings, Springer Science Business Media, 2002.

K. Nagata, F. Nemenzo, H. Wada, The number of self-dual codes over Zp3 , Des. Codes Cryptogr. 50 (2009) 291–303.

Y. Niu, Q. Yue, Y. Wu, L. Hu, Hermitian self-dual, MDS, and generalized Reed-Solomon codes, IEEE Commun. Lett. 23 (2019) 781–784.

G. H. Norton, A. Sˇalˇagean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Info. Theory 46 (2000) 1060–1067.

G. H. Norton, A. Sˇalˇagean, On the structure of linear and cyclic codes over a finite chain ring, AAECC 10 (2000) 489–506.

D. S. Passman, The Algebraic Structure of Group Rings, Wiley, New York, 1977.

V. Pless, On the uniqueness of the Golay codes, J. Comb. Theory 5 (1968) 215–228.

E. M. Rains, N. J. A. Sloane, Self-dual codes. In: Handbook of Coding Theory, pp. 177–294. North-Holland, Amsterdam 1998.

B. S. Rajan, M. U. Siddiqi, Transform domain characterization of abelian codes, IEEE Trans. Inform. Theory 38 (1992) 1817–1821.

H. Tong, X. Wang, New MDS Euclidean and Hermitian self-dual codes over finite fields, Advances in Pure Mathematics 7 (2017) 325–333.