Efficient Decoding Algorithm for Binary Quadratic Residue Codes Using Reduced Permutation Sets

Abstract

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

A new efficient decoder of linear block codes based on ensemble learning models: case of boosting

“…Khebbou et al [21] have adapted a polar code decoding technique in favor of the extended Golay code. Boualame et al [22] have proposed a decoder that uses a condensed set of permutations drawn from the huge automorphism group of QR codes to rectify t or fewer incorrect bits in the received word. In the realm of machine learning, computers have the ability to learn and evolve through experience, without the need for explicit programming [23].…”