Generalized Minimum Distance (GMD) decoding is a well-known soft-decision decoding for linear codes. Previous research on GMD decoding focused mainly on unquantized AWGN channels with BPSK signaling for binary linear codes. In this paper, a study on the design of a 4-level uniform quantizer for GMD decoding is given. In addition, an extended version of a GMD decoding algorithm for a 4-level quantizer is proposed, and the effectiveness of the proposed decoding is shown by simulation.

In 1996, Sipser and Spielman [12] constructed a family of linear-time decodable asymptotically good codes called expander codes. Recently, Barg and Zemor [2] gave a modified construction of expander codes, which greatly improves the code parameters. In this paper we present a new simple algebraic decoding algorithm for the modified expander codes of Barg and Zemor, and give a Justesen-type construction of linear-time decodable asymptotically good binary linear codes that meet the Zyablov bound.

For binary linear block codes, we introduce "multiple GMD decoding algorithm. " In this algorithm, GMD-like decoding is iterated around a few appropriately selected search centers. The original GMD decoding by Forney is a GMD-like decoding around the hard-decision sequence. Compared with the original GMD decoding, this decoding algorithm provides better error performance with moderate increment of iteration numbers. To reduce the number of iterations, we derive new effective sufficient conditions on the optimality of decoded codewords.

A unified algorithm is presented for solving key equations for decoding alternant codes. The algorithm can be applied to various decoding techniques, including bounded distance decoding, generalized minimum distance decoding, Chase decoding, etc.

In this paper, we provide an efficient algorithm for GMD (Generalized Minimum Distance) decoding of BCH codes over q-valued logic, when q is pl (p: prime number, l: positive integer). An algebraic errors-and-erasures decoding procedure is required to execute only one time, whereas in a conventional GMD decoding at mostd/2algebraic decodings are necessary, where d is the design distance of the code. In our algorithm, Welch-Berlekamp's iterative method is efficiently employed to reduce the number of algebraic decoding procedures. We also show a method for hardware implementation of this GMD decoding based on q-valued logic.