Dualilty Preserving Gray Maps: (Pseudo) Self-dual Bases and Symmetric Bases

Given a finite ring A which is a free left module over a subring R of A, two types of R-bases are defined which in turn are used to define duality preserving maps from codes over A to codes over R. The first type, pseudo-self-dual bases, are a generalization of trace orthogonal bases for fields. The second are called symmetric bases. Both types are illustrated with skew cyclic codes which are codes that are A-submodules of the skew polynomial ring A[X;�]/hX n − 1i (the classical cyclic codes are the case when � = id). When A is commutative, there exists criteria for a skew cyclic code over A to be self-dual. With this criteria and a duality preserving map, many self-dual codes over the subring R can easily be found. In this fashion, numerous examples are given, some of which are not chain or serial rings.

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

[2]  Felix Ulmer,et al.  Self-dual skew codes and factorization of skew polynomials , 2014, J. Symb. Comput..

[3]  Abraham Lempel,et al.  Factorization of Symmetric Matrices and Trace-Orthogonal Bases in Finite Fields , 1980, SIAM J. Comput..

[4]  Christine Bachoc,et al.  Applications of Coding Theory to the Construction of Modular Lattices , 1997, J. Comb. Theory A.

[5]  Suat Karadeniz,et al.  Linear codes over F 2 + uF 2 + vF 2 + uvF 2 . , 2010 .

[6]  Koichi Betsumiya,et al.  Type II codes over F2m+uF2m , 2004, Discret. Math..

[7]  N. J. A. Sloane,et al.  On the Classification and Enumeration of Self-Dual Codes , 1975, J. Comb. Theory, Ser. A.

[8]  Patrick Solé,et al.  Type II Codes OverF4 + uF4 , 2001, Eur. J. Comb..

[9]  Pierre-Louis Cayrel,et al.  Quasi-cyclic codes as codes over rings of matrices , 2010, Finite Fields Their Appl..

[10]  R. Alfaro,et al.  Constructing self-dual codes over $$\mathbb{F }_q[u]/(u^t)$$Fq[u]/(ut) , 2015, Des. Codes Cryptogr..

[11]  Koichi Betsumiya,et al.  Type II codes over F 2 m +u F 2 m , 2004 .

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

[13]  W. Cary Huffman,et al.  On the classification and enumeration of self-dual codes , 2005, Finite Fields Their Appl..

[14]  E. Rains,et al.  3-Colored 5-Designs and Z4-Codes , 2000 .

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

[16]  Judy L. Walker.  Algebraic Geometric Codes over Rings , 1999 .

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

[18]  Patrick Solé,et al.  Type II Codes Over F 4 + uF 4 . , 2001 .

[19]  Patrick Solé,et al.  Cyclic codes over M2(F2) , 2013, J. Frankl. Inst..

[20]  Suat Karadeniz,et al.  Linear codes over F2+uF2+vF2+uvF2 , 2010, Des. Codes Cryptogr..

[21]  Edgar Martínez-Moro,et al.  Linear codes over Z4[x]/〈x² + 2x〉 , 2015 .

[22]  Felix Ulmer,et al.  A Note on the Dual Codes of Module Skew Codes , 2011, IMACC.

[23]  Wang Yon.  Type II Codes Over F_2+uF_2+ … +u~kF_2 , 2015 .

[24]  Patrick Solé,et al.  EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES , 2006 .

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

[26]  Taher Abualrub,et al.  On θ-cyclic codes over F2 + vF2 , 2012 .

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

[28]  Marcus Greferath,et al.  Cyclic codes over finite rings , 1997, Discret. Math..

[29]  Jay A. Wood.  Duality for modules over finite rings and applications to coding theory , 1999 .