# Constructions of MDS Convolutional Codes Using Superregular Matrices

### Abstract

Maximum distance separable convolutional codes are the codes that present best performance in error correction among all convolutional codes with certain rate and degree. In this paper, we show that taking the constant matrix coefficients of a polynomial matrix as submatrices of a superregular matrix, we obtain a column reduced generator matrix of an MDS convolutional code with a certain rate and a certain degree. We then present two novel constructions that fulfill these conditions by considering two types of superregular matrices.

### References

P. J. Almeida, D. Napp, R. Pinto, Superregular matrices and applications to convolutional codes, Linear Algebra Appl. 499 (2016) 1–25.

J. Climent, D. Napp, C. Perea, R. Pinto, A construction of MDS 2D convolutional codes of rate 1/n based on superregular matrices, Linear Algebra Appl. 437(3) (2012) 766–780.

J. Climent, D. Napp, C. Perea, R. Pinto, Maximum distance seperable 2D convolutional codes, IEEE Trans. Inform. Theory 62(2) (2016) 669–680.

G. Forney, Convolutional codes I: Algebraic structure, IEEE Transactions on Information Theory, 16(6) (1970) 720–738. Correction, Ibid., IT-17, (1971) 360.

H. Gluesing–Luerssen, B. Langfeld, A class of one–dimensional MDS convolutional codes, J. Algebra Appl. 5(4) (2006) 505–520.

H. Gluesing–Luerssen, J. Rosenthal, R. Smarandache, Strongly–MDS convolutional codes, IEEE Trans. Inform. Theory 52(2) (2006) 584–598.

R. Hutchinson, J. Rosenthal, R. Smarandache, Convolutional codes with maximum distance profile, Systems & Control Letters 54 (2005) 53–63.

J. Justesen, An algebraic construction of rate 1/ν convolutional codes, IEEE Trans. Inform. Theory 21(5) (1975) 577–580.

T. Kailath, Linear Systems, Englewood Cliffs, N.J.: Prentice Hall, 1980.

J. Lieb, Complete MDP convolutional codes, J. Algebra Appl. 18(6) (2019) 1950105 (13 pages).

F. J. MacWilliams, N. J. A. Sloane, The Theory of Error–Correcting Codes, 6th ed. Amsterdam, The Netherlands: North–Holland, 1988.

J. Rosenthal, R. Smarandache, Maximum distance separable convolutional codes, Appl. Algebra Engrg. Comm. Comput. 10(1) (1999) 15–32.

R. Roth, A. Lempel, On MDS codes via Cauchy matrices, IEEE Trans. Inform. Theory 35(6) (1989) 1314–1319.

R. Smarandache, H. Gluesing–Luerssen, J. Rosenthal, Constructions for MDS–convolutional codes, IEEE Trans. Inform. Theory 47(5) (2001) 2045–2049.

R. Smarandache, J. Rosenthal, A state space approach for constructing MDS rate 1/n convolutional codes, Proceedings of the 1998 IEEE Information Theory Workshop on Information Theory, 116–117.