Algebraic Decoding for Quadratic Residue Codes by Using Newton Identities

本研究的主要目的是寻找二次剩余(QR)码的译码算法，包括众所周知的(23，12，7)Golay码、(31，16，7)QR码和(41，21，9)码。其关键思想是通过系统地使用与码子集相关联的牛顿恒等式来寻找错误定位多项式。研究的重点是长度为8m+/-1的二进制QR码。所发展的技术将PI最近针对(31，16，7)QR码和Elia针对(23，12，7)Golay码提出的代数译码算法扩展到更一般的QR码。这个扩展的第一个新示例是(41，21，9)二维码。期望这项工作和这里开发的代数方法一般可以应用于QR码的整个集合以及诸如BCH和Reed-Solomon码的其他码。

The primary objectives of this research are to find decoding algorithms for the Quadratic Residue (QR) Codes, including the well known (23,12,7) Golay code, the (31,16,7) QR code and the (41,21,9) code. The key idea is to find the error locator polynomial by a systematic use of the Newton identities associated with the code syndromes. The study focusses on the binary QR codes which have length 8m+/- 1. The techniques developed extend the algebraic decoding algorithm found recently by the PI for the (31,16,7) QR code and by Elia for the (23,12,7) Golay code to more general QR codes. The first new example of this extension is the (41,21,9) QR code. Is expected that this work and the algebraic methods developed here can apply generally to the entire set of QR codes and to other codes such as the BCH and Reed-Solomon codes.