LCD and Self-Orthogonal Group Codes in a Finite Abelian $p$ -Group Algebra

Let <inline-formula> <tex-math notation="LaTeX">$\Bbb F_{q}$ </tex-math></inline-formula> be a finite field with <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> elements and <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> be a prime with <inline-formula> <tex-math notation="LaTeX">$\gcd (p,q)=1$ </tex-math></inline-formula>. Let <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> be a finite abelian <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>-group and <inline-formula> <tex-math notation="LaTeX">$\Bbb F_{q}(G)$ </tex-math></inline-formula> be a group algebra. In this paper, we find all primitive idempotents and minimal abelian group codes in the group algebra <inline-formula> <tex-math notation="LaTeX">$\Bbb F_{q}(G)$ </tex-math></inline-formula>. Furthermore, we give all LCD abelian codes (linear code with complementary dual) and self-orthogonal abelian codes of <inline-formula> <tex-math notation="LaTeX">$\Bbb F_{q}(G)$ </tex-math></inline-formula>.

[1]  Chengju Li,et al.  The minimum Hamming distances of irreducible cyclic codes , 2014, Finite Fields Their Appl..

[2]  Patanee Udomkavanich,et al.  Characterization and enumeration of complementary dual abelian codes , 2017, ArXiv.

[3]  F. J. Mac Williams.  Binary codes which are ideals in the group algebra of an abelian group , 1970 .

[4]  Cunsheng Ding,et al.  Two Families of LCD BCH Codes , 2016, IEEE Transactions on Information Theory.

[5]  Sudhir Batra,et al.  Some cyclic codes of length 2pn , 2011, Des. Codes Cryptogr..

[6]  Bocong Chen,et al.  New Constructions of MDS Codes With Complementary Duals , 2017, IEEE Transactions on Information Theory.

[7]  Chengju Li,et al.  Irreducible cyclic codes of length 4pn and 8pn , 2015, Finite Fields Their Appl..

[8]  F. E. Brochero Martínez,et al.  Explicit idempotents of finite group algebras , 2014, Finite Fields Their Appl..

[9]  Qin Yue,et al.  The primitive idempotents and weight distributions of irreducible constacyclic codes , 2018, Des. Codes Cryptogr..

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

[11]  Hongwei Liu,et al.  Abelian Codes in Principal Ideal Group Algebras , 2013, IEEE Transactions on Information Theory.

[12]  Gabriele Nebe,et al.  Self-dual codes and invariant theory , 2009, Algebraic Aspects of Digital Communications.

[13]  Raul Antonio Ferraz,et al.  Idempotents in group algebras and minimal abelian codes , 2007, Finite Fields Their Appl..

[14]  Manju Pruthi,et al.  Minimal Cyclic Codes of Length 2pn , 1999 .

[15]  Cunsheng Ding,et al.  LCD Cyclic Codes Over Finite Fields , 2017, IEEE Transactions on Information Theory.

[16]  Hao Chen,et al.  Secure Computation from Random Error Correcting Codes , 2007, EUROCRYPT.

[17]  Gurmeet K. Bakshi,et al.  Minimal cyclic codes of length pnq , 2003 .

[18]  Mohammad Umar Siddiqi,et al.  Transform domain characterization of Abelian codes , 1992, IEEE Trans. Inf. Theory.

[19]  Javier de la Cruz,et al.  On group codes with complementary duals , 2018, Des. Codes Cryptogr..

[20]  Hongwei Liu,et al.  A class of minimal cyclic codes over finite fields , 2015, Des. Codes Cryptogr..

[21]  Manju Pruthi.  Cyclic codes of length 2m , 2001 .

[22]  Sihem Mesnager,et al.  Euclidean and Hermitian LCD MDS codes , 2017, Des. Codes Cryptogr..

[23]  Cunsheng Ding,et al.  Split group codes , 2000, IEEE Trans. Inf. Theory.