Wide-Sense Time-Invariant Block Codes in Their Generator Matrix
A wide-sense time-invariant block code is characterized by more than one generator polynomial periodically repeating in its generator matrix. Interleaved multiplication enables us to construct a code word. A quasi-cyclic code can be considered as a generalization of a cyclic code, with a characteristic block length, always integer multiple of the period. These codes are presented in a reordered form, traditionally adopted. An interpretation of known block codes with composite block length as quasi-cyclic codes is provided. They can be set in correspondence with cyclic or pseudo-cyclic codes in a non-binary alphabet. Encoder circuits for quasi-cyclic codes are discussed, always adopting the generator matrix model. Their evolution is described in a non-binary state diagram. Also for quasi-cyclic codes, the operations of code shortening, code puncturation, and code lengthening are possible, with reference to either the original form or a properly reordered form. A first introduction to array codes is given. Permutation circulants are adopted as very efficient tools for performing an interleaver function. Modified lengthened quasi-cyclic codes show the typical characteristics of wide-sense time-invariant convolutional codes. The concept of subcodes is extended to the case of quasi-cyclic codes. Finally, some trellises of quasi-cyclic codes are discussed, especially when they allow a simplified computational complexity in soft-decision decoding through a proper partitioning.
KeywordsGenerator Matrix Block Code LDPC Code Code Word Cyclic Code
- Baldi M et al (2012) On the generator matrix of array codes. In: Proceedings of Softcom 2012, Split, CroatiaGoogle Scholar
- Calderbank AR, Forney GD, Vardy A (1998) Minimal tail-biting trellises: the Golay code and more. In: Proceedings of ISIT 1998, Cambridge, USA, p 255Google Scholar
- Fan JL (2000) Array codes as low-density parity check codes. In: Proceedings of 2nd international symposium turbo codes, Brest, France, pp 543–546Google Scholar
- Kasami T, Takata T, Fujiwara T, Lin S (1990) Trellis diagram construction for some BCH codes. In: Proceedings of IEEE International Symposium Information Theory and Application, HonoluluGoogle Scholar
- Lin S, Costello DJ (2004) Error control coding. Pearson, Prentice-Hall, Upper Saddle RiverGoogle Scholar