Left dihedral codes over Galois rings GR(p2, m)

Abstract Let D 2 n = 〈 x , y ∣ x n = 1 , y 2 = 1 , y x y = x − 1 〉 be a dihedral group, and R = GR ( p 2 , m ) be a Galois ring of characteristic p 2 and cardinality p 2 m where p is a prime. Left ideals of the group ring R [ D 2 n ] are called left dihedral codes over R of length 2 n , and abbreviated as left D 2 n -codes over R . Let gcd ( n , p ) = 1 in this paper. Then any left D 2 n -code over R is uniquely decomposed into a direct sum of concatenated codes with inner codes A i and outer codes C i , where A i is a cyclic code over R of length n and C i is a skew cyclic code of length 2 over a Galois ring or principal ideal ring extension of R . Specifically, a generator matrix and basic parameters for each outer code C i are given. A formula to count the number of these codes is obtained and the dual code for each left D 2 n -code is determined. Moreover, all self-dual left D 2 n -codes and self-orthogonal left D 2 n -codes over R are presented.

[1]  Ana Salagean,et al.  Repeated-root cyclic and negacyclic codes over a finite chain ring , 2006, Discret. Appl. Math..

[2]  Patanee Udomkavanich,et al.  Skew constacyclic codes over finite chain rings , 2010, Adv. Math. Commun..

[3]  Yonglin Cao,et al.  Concatenated structure of left dihedral codes , 2016, Finite Fields Their Appl..

[4]  A. Nechaev,et al.  Kerdock code in a cyclic form , 1989 .

[5]  Gabriela Olteanu,et al.  Construction of minimal non-abelian left group codes , 2013, Des. Codes Cryptogr..

[6]  Patrick Solé,et al.  Skew constacyclic codes over Galois rings , 2008, Adv. Math. Commun..

[7]  Sergio R. López-Permouth,et al.  Cyclic and negacyclic codes over finite chain rings , 2004, IEEE Transactions on Information Theory.

[8]  Jie Cui,et al.  Quaternary 1-generator quasi-cyclic codes , 2011, Des. Codes Cryptogr..

[9]  Sucheta Dutt,et al.  Cyclic Codes over Galois Rings , 2016, CALDAM.

[10]  Shixin Zhu,et al.  Negacyclic self-dual codes over finite chain rings , 2012, Des. Codes Cryptogr..

[11]  Sanjoy K. Mitter,et al.  Some randomized code constructions from group actions , 2006, IEEE Transactions on Information Theory.

[12]  N. J. A. Sloane,et al.  A linear construction for certain Kerdock and Preparata codes , 1993, ArXiv.

[13]  Eimear Byrne,et al.  Decoding a class of Lee metric codes over a Galois ring , 2002, IEEE Trans. Inf. Theory.

[14]  Raul Antonio Ferraz,et al.  Semisimple group codes and dihedral codes , 2009 .

[15]  D. Passman,et al.  The algebraic structure of group rings , 1977 .

[16]  Felix Ulmer,et al.  Skew-cyclic codes , 2006, Applicable Algebra in Engineering, Communication and Computing.

[17]  Eimear Byrne,et al.  Lifting Decoding Schemes over a Galois Ring , 2001, AAECC.

[18]  Z. Wan.  Lectures on Finite Fields and Galois Rings , 2003 .

[19]  Yonglin Cao,et al.  Generalized quasi-cyclic codes over Galois rings: structural properties and enumeration , 2011, Applicable Algebra in Engineering, Communication and Computing.

[20]  Patanee Udomkavanich,et al.  The Gray Image of Codes over Finite Chain Rings , 2009, ArXiv.

[21]  Maheshanand Bhaintwal.  Skew quasi-cyclic codes over Galois rings , 2012, Des. Codes Cryptogr..

[22]  Wolfgang Willems,et al.  Self-Dual Doubly Even $2$-Quasi-Cyclic Transitive Codes Are Asymptotically Good , 2007, IEEE Transactions on Information Theory.

[23]  Eimear Byrne,et al.  Hamming metric decoding of alternant codes over Galois rings , 2002, IEEE Trans. Inf. Theory.

[24]  N. J. A. Sloane,et al.  The Z4-linearity of Kerdock, Preparata, Goethals, and related codes , 1994, IEEE Trans. Inf. Theory.

[25]  Nicolas Sendrier,et al.  On the Concatenated Structure of a Linear Code , 1998, Applicable Algebra in Engineering, Communication and Computing.

[26]  Graham H. Norton,et al.  On the Structure of Linear and Cyclic Codes over a Finite Chain Ring , 2000, Applicable Algebra in Engineering, Communication and Computing.

[27]  Ian McLoughlin.  A group ring construction of the [48,24,12] type II linear block code , 2012, Des. Codes Cryptogr..

[28]  Dwijendra K. Ray-Chaudhuri,et al.  Quasi-cyclic codes over Z4 and some new binary codes , 2002, IEEE Trans. Inf. Theory.

[29]  Felix Ulmer,et al.  Coding with skew polynomial rings , 2009, J. Symb. Comput..

[30]  Yonglin Cao.  1-generator quasi-cyclic codes over finite chain rings , 2012, Applicable Algebra in Engineering, Communication and Computing.

[31]  Maheshanand Bhaintwal,et al.  On quasi-cyclic codes over $${\mathbb{Z}_q}$$ , 2009, Applicable Algebra in Engineering, Communication and Computing.

[32]  A. Sălăgean,et al.  On the stucture of linear and cyclic codes over finite chain rings , 2018 .

[33]  H. Q. Dinh,et al.  Complete Distances of All Negacyclic Codes of Length $2^{s}$ Over $\BBZ _{2^{a}}$ , 2007, IEEE Transactions on Information Theory.

[34]  Fabio Enrique Brochero Martínez,et al.  Structure of finite dihedral group algebra , 2014, Finite Fields Their Appl..

[35]  Yonglin Cao.  On constacyclic codes over finite chain rings , 2013, Finite Fields Their Appl..

[36]  Felix Ulmer,et al.  Codes as Modules over Skew Polynomial Rings , 2009, IMACC.

[38]  Han Mao Kiah,et al.  Cyclic codes over GR(p2, m) of length pk , 2008, Finite Fields Their Appl..