An Improved Decoding Algorithm of the (71, 36, 11) Quadratic Residue Code Without Determining Unknown Syndromes

Abstract

In this paper, a new algebraic method to decode the (71, 36, 11) QR code up to five errors is proposed. It completely avoids computing the unknown syndromes, and uses the previous scheme of decoding this QR code up to three errors, but corrects four and five errors with a new different method. In the four-error case, the new algorithm directly determines the coefficients of the error-locator polynomial by eliminating unknown syndromes in Newton identities. Subsequently, the shift-search algorithm can be utilized to decode the fifth error and the concept of bit reliability is also introduced to accelerate the decoding process. In other words, a weight-five-error pattern can be decoded in terms of the four-error case by inverting an incorrect bit of the received word in ascending order of reliability. Particularly, a threshold parameter $\gamma$ can be preset to limit the number of inverting bits one by one, and a corresponding upper bound of the probability that decoding fails is derived. Finally, simulation and analysis show that the proposed new decoding algorithm for the abovementioned QR code not only significantly reduces the decoding complexity in terms of CPU time but also saves a lot of memory while maintaining the same error-rate performance. Additionally, the introduction of $\gamma$ achieves a better tradeoff between the decoding performance and the computational complexity.