Superregular matrices and their importance in convolutional coding theory

Ryan Hutchinson, Jochen Trumpf, Joachim Rosenthal

Superregular lower triangular Toeplitz matrices over a finite field were introduced in [1] as a tool for constructing a certain class of convolutional codes. It is desirable to obtain an algebraic construction of superregular lower Toeplitz matrices that leads to codes for which an algebraic decoding algorithm is possible. Unfortunately, we are not able to offer such a construction at this time. We present instead an upper bound on the field size required for the existence of a superregular lower triangular Toeplitz matrix of a given size, as well as methods for constructing new superregular lower triangular Toeplitz matrices from an existing one. The bound is given in terms of Catalan numbers.

[1] H. Gluesing-Luerssen, J. Rosenthal and R. Smarandache. Strongly MDS convolutional codes. March 2003.