Low complexity encoder for generalized quasi-cyclic codes coming from finite geometries

Authors:
Vo Tam Van;Hajime Matsui;Seiichi Mita
Affiliations:
Dept. Electronics and Information Science, Toyota Technological Institute, Tenpaku, Nagoya, Japan;Dept. Electronics and Information Science, Toyota Technological Institute, Tenpaku, Nagoya, Japan;Dept. Electronics and Information Science, Toyota Technological Institute, Tenpaku, Nagoya, Japan
Venue:
ICC'09 Proceedings of the 2009 IEEE international conference on Communications
Year:
2009

Citing 6
Cited 1

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Some Classes of Quasi-Cyclic LDPC Codes: Properties and Efficient Encoding Method*This paper was presented in part at the 26th Symposium on Information Theory and Its Applications, Higashiura, Hyogo, Japan, December 15--18, 2003 and at IEEE International Symposium on Information Theory, Chicago, IL USA, June 27--July 2, 2004.

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Structural properties and enumeration of 1-generator generalized quasi-cyclic codes

Designs, Codes and Cryptography

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Abstract

We define generalized quasi-cyclic (GQC) codes as linear codes with nontrivial automorphism groups. Therefore, GQC codes, unlike quasi-cyclic codes, can include many important codes such as Hermitian and projective geometry (PG) codes; this capability is important in practical applications. Further, we propose the echelon canonical form algorithm for computing Gröbner bases from their parity check matrices. Consequently, by applying Gröbner base theory, GQC codes can be systematically encoded and implemented with simple feedback shift registers. Our algorithm is based on Gaussian elimination and requires a sufficiently small number of finite-field operations, which is related to the third power of code-length. In order to demonstrate our encoder's efficiency, we prove that the number of circuit elements in the encoder architecture is proportional to the code-length for finite geometry (FG) LDPC codes (a class of GQC codes). We show that the hardware complexity of a serial-in-serial-out encoder architecture for FG-LDPC codes is related to the linear order of the code-length; less than 2n adder and 2n memory elements are required to encode a binary codeword of length n.