A new class of superregular matrices and MDP convolutional codes

This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a suficiently large finite field F. Such construction works for any given choice of characteristic of the field F and code parameters (n; k; d) such that (n-k)|d. Finally, we discuss the size of F needed so that the proposed matrices are superregular.

[1]  Joachim Rosenthal,et al.  Constructions of MDS-convolutional codes , 2001, IEEE Trans. Inf. Theory.

[2]  Ryan Hutchinson.  The Existence of Strongly MDS Convolutional Codes , 2008, SIAM J. Control. Optim..

[3]  Rolf Johannesson,et al.  Fundamentals of Convolutional Coding , 1999 .

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  Virtudes Tomás Estevan.  Complete-mdp convolutional codes over the erasure channel , 2011 .

[6]  Joachim Rosenthal,et al.  Strongly-MDS convolutional codes , 2003, IEEE Transactions on Information Theory.

[7]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[8]  Ron M. Roth,et al.  On MDS codes via Cauchy matrices , 1989, IEEE Trans. Inf. Theory.

[9]  Jochen Trumpf,et al.  On superregular matrices and MDP convolutional codes , 2008 .

[10]  Joachim Rosenthal,et al.  Decoding of MDP convolutional codes over the erasure channel , 2009, 2009 IEEE International Symposium on Information Theory.

[11]  Ron M. Roth,et al.  On generator matrices of MDS codes , 1985, IEEE Trans. Inf. Theory.

[12]  R. Tennant.  Algebra , 1941, Nature.

[13]  Joachim Rosenthal,et al.  Maximum Distance Separable Convolutional Codes , 1999, Applicable Algebra in Engineering, Communication and Computing.

[14]  Joachim Rosenthal,et al.  Convolutional codes with maximum distance profile , 2003, Syst. Control. Lett..

[15]  José Ignacio Iglesias Curto,et al.  Convolutional Goppa codes , 2003, IEEE Transactions on Information Theory.