Polycyclic codes as invariant subspaces

Abstract Polycyclic codes are a powerful generalization of cyclic and constacyclic codes. Their algebraic structure is studied here by the theory of invariant subspaces from linear algebra. As an application, a bound on the minimum distance of these codes is derived which outperforms, in some cases, the natural analogue of the BCH bound.

[1]  W. Greub.  Linear Algebra , 1981 .

[2]  Edgar Martínez-Moro,et al.  On polycyclic codes over a finite chain ring , 2018, Adv. Math. Commun..

[3]  Minjia Shi,et al.  Quasi-twisted codes with constacyclic constituent codes , 2016, Finite Fields Their Appl..

[4]  Patrick Solé,et al.  On the additive cyclic structure of quasi-cyclic codes , 2018, Discret. Math..

[5]  Patrick Solé,et al.  On the duality and the direction of polycyclic codes , 2016, Adv. Math. Commun..

[6]  Minjia Shi,et al.  A modified Gilbert-Varshamov bound for self-dual quasi-twisted codes of index four , 2020, Finite Fields Their Appl..

[7]  Sergio R. López-Permouth,et al.  Dual generalizations of the concept of cyclicity of codes , 2009, Adv. Math. Commun..

[8]  Sergio R. López-Permouth,et al.  Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes , 2012, Finite Fields Their Appl..

[9]  Edgar Martínez-Moro,et al.  On repeated-root multivariable codes over a finite chain ring , 2007, Des. Codes Cryptogr..

[10]  D. Chillag.  Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes , 1995 .

[11]  Gennian Ge,et al.  Pseudo-cyclic Codes and the Construction of Quantum MDS Codes , 2016, IEEE Transactions on Information Theory.

[12]  W. W. Peterson,et al.  Error-Correcting Codes. , 1962 .

[13]  A. J. van Zanten,et al.  Constacyclic codes as invariant subspaces , 2009 .