Construction of Unit-Memory MDS Convolutional Codes

Maximum-distance separable (MDS) convolutional codes form an optimal family of convolutional codes, the study of which is of great importance. There are very few general algebraic constructions of MDS convolutional codes. In this paper, we construct a large family of unit-memory MDS convolutional codes over $\F$ with flexible parameters. Compared with previous works, the field size $q$ required to define these codes is much smaller. The construction also leads to many new strongly-MDS convolutional codes, an important subclass of MDS convolutional codes proposed and studied in \cite{GL2}. Many examples are presented at the end of the paper.

[1]  Christoforos N. Hadjicostis,et al.  Encoded dynamics for fault tolerance in linear finite-state machines , 2002, IEEE Trans. Autom. Control..

[2]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[3]  Guanghui Zhang,et al.  A Construction of MDS Quantum Convolutional Codes , 2014, ArXiv.

[4]  Robert M. Gray,et al.  Coding for noisy channels , 2011 .

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

[6]  Jianzhang Chen,et al.  Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes , 2015 .

[7]  Philippe Piret,et al.  Convolutional Codes: An Algebraic Approach , 1988 .

[8]  Christoforos N. Hadjicostis,et al.  Non-Concurrent Error Detection and Correction in Fault-Tolerant Discrete-Time LTI Dynamic Systems , 2003 .

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

[10]  Andrew J. Viterbi,et al.  Error bounds for convolutional codes and an asymptotically optimum decoding algorithm , 1967, IEEE Trans. Inf. Theory.

[11]  Pradeep Kiran Sarvepalli,et al.  Quantum Convolutional BCH Codes , 2007, 2007 10th Canadian Workshop on Information Theory (CWIT).

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

[13]  Giuliano G. La Guardia,et al.  On Classical and Quantum MDS-Convolutional BCH Codes , 2012, IEEE Transactions on Information Theory.

[14]  Christoforos N. Hadjicostis,et al.  Finite-state machine embeddings for nonconcurrent error detection and identification , 2005, IEEE Transactions on Automatic Control.

[15]  Lin-nan Lee,et al.  Short unit-memory byte-oriented binary convolutional codes having maximal free distance (Corresp.) , 1976, IEEE Trans. Inf. Theory.

[16]  Jørn Justesen,et al.  Bounds on distances and error exponents of unit memory codes , 1983, IEEE Trans. Inf. Theory.

[17]  Yuansheng Tang,et al.  Quantum convolutional codes derived from constacyclic codes , 2014 .

[18]  Giuliano G. La Guardia,et al.  On Negacyclic MDS-Convolutional Codes , 2013, ArXiv.

[19]  Giuliano G. La Guardia.  On nonbinary quantum convolutional BCH codes , 2012, Quantum Inf. Comput..

[20]  W. J. Ebel.  A directed search approach for unit-memory convolutional codes , 1996, IEEE Trans. Inf. Theory.

[21]  H. Gluesing-Luerssen,et al.  Construction results for MDS-convolutional codes , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).