A Low-Complexity Step-by-Step Decoding Algorithm for Binary BCH Codes

Abstract

A low-complexity step-by-step decoding algorithm for terror correcting binary Bose-Chaudhuri-Hocquenghem (BCH) codes is proposed. Using logical analysis, we obtained a simple rule which can directly determine whether a bit in the received word is correct. The computational complexity of this decoder is less than the conventional step-by-step decoding algorithm, since it reduces at least half of the matrix computations and the most complex element in the conventional step-by-step decoder is the "matrix-computing" element.

Low-Complexity Triple-Error-Correcting Parallel BCH Decoder

“…As mentioned, S i (i = 1, 3,5) are required in triple-error-correcting m-SBS algorithmbased BCH decoding to calculate the syndrome. Because the SC receives the parallelized codeword, a parallel SC cell is required, as shown in Fig.…”

“…Another design method for BCH codes, which has a decoding procedure entirely different from the BM algorithm, is the Peterson algorithm [3] and the step-bystep (SBS) decoding algorithm [4][5][6]. The decoding procedure of the SBS decoding algorithm also consists of three steps, as follows: First, calculate the syndrome values S i (i=1, 2, ..., 2t) from the received codeword.…”

“…This is the main weakness of the SBS decoding algorithm, which makes it less competitive than the BM algorithm [6]. To reduce the complexity of the SBS decoding algorithm, previous studies [5] and [6] have proposed a low-complexity SBS decoding algorithm to simplify the decoding procedure and increase the decoding speed. However, for every received symbol, at least one determinant of a v × v matrix, where v is the number of errors, still has to be calculated.…”

Decoding the (23, 12, 7) Binary Golay Code

“…By repeating this procedure for all information positions of the error pattern can be found. By applying theorem 3 in [7], a modified step-by-step decoding algorithm for the (23, 12, 7) binary Golay code is proposed. The algorithm follows the same idea as that proposed in [6], but further reduces the amount of the calculations of variables (i.e., T 3 , T 9 , and F) by means of logical analysis.…”

“…The algorithm follows the same idea as that proposed in [6], but further reduces the amount of the calculations of variables (i.e., T 3 , T 9 , and F) by means of logical analysis. Based on theorem 3 in [7], the determination whether a received bit is erroneous in the method as proposed in [6] can be further simplified into a simple equation. Based on Theorem 1, a modified step-by-step decoding algorithm for the (23, 12, 7) binary Golay code can be described as follows: 1) Calculate the original syndromes S i (i=1, 3,9).…”