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Studying the locator polynomials of minimum weight codewords of BCH codes

机译：研究BCH码最小权重码字的定位多项式

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摘要

Primitive binary cyclic codes of length n=2/sup m/ are considered. A BCH code with designed distance delta is denoted B(n, delta ). A BCH code is always a narrow-sense BCH code. A codeword is identified with its locator polynomial, whose coefficients are the symmetric functions of the locators. The definition of the code by its zeros-set involves some properties for the power sums of the locators. Moreover, the symmetric functions and the power sums of the locators are related to Newton's identities. An algebraic point of view is presented in order to prove or disprove the existence of words of a given weight in a code. The principal result is the true minimum distance of some BCH codes of length 255 and 511. which were not known. The minimum weight codewords of the codes B(n2/sup h/-1) are studied. It is proved that the set of the minimum weight codewords of the BCH code B(n,2/sup m-2/-1) equals the set of the minimum weight codewords of the punctured Reed-Muller code of length n and order 2, for any m.
机译：考虑长度为n = 2 / sup m /的原始二进制循环码。具有设计距离增量的BCH码表示为B（n，delta）。 BCH码始终是狭义BCH码。用其定位多项式识别码字，其系数是定位器的对称函数。代码的零位定义涉及定位器的功率和的某些属性。此外，定位器的对称函数和幂和与牛顿身份有关。提出了代数观点，以证明或反对代码中给定权重的单词的存在。主要结果是一些未知的长度为255和511的BCH码的真实最小距离。研究了码B（n2 / sup h / -1）的最小权重码字。证明BCH码B（n，2 / sup m-2 / -1）的最小权重码字的集合等于长度为n且阶数为2的穿孔Reed-Muller码的最小权重码字的集合，对于任何米。

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