Algebraic Decoding for Quadratic Residue Codes by Using Newton Identities

本研究的主要目的是找到解码 算法的二次剩余（QR）码，包括井 已知的（23，12，7）Golay码、（31，16，7）QR码和（41，21，9） 代码. 关键思想是通过一个 系统地使用与代码相关的牛顿恒等式 综合征 研究的对象是长度为8 m +/-的二进制QR码 1.所开发的技术扩展了代数解码算法 PI最近发现了（31，16，7）QR码，Elia发现了 （23，12，7）Golay码到更一般的QR码。 第一个新 这种扩展的示例是（41，21，9）QR码。 预计 这项工作和这里开发的代数方法可以应用于 通常是整个QR码集和其他码， BCH码和Reed-Solomon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.