Cyclic codes from irreducible polynomials for correction of multiple errors
Cyclic codes from irreducible polynomials for correction of multiple errors
复制标题
用于纠正多个错误的不可约多项式的循环码
DOI：
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发表时间：
1962
期刊：
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通讯作者：
L. Zetterberg
中科院分区：
文献类型：
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作者：
L. Zetterberg
研究了一类用于纠正多重错误的移位寄存器码。定义矩阵将满足系数为0或1的不可约多项式方程。根据误差环对误差进行了分类，并提出了一个简单的方法来确定误差环是否是可区分的。为了便于分析，引入了另一种错误分类，使错误位数（权重）不变。已经研究了许多特定的代码，用于纠正错误突发或纠正某些权重的所有多个错误。脉冲串长度2、3、4、5和6被尝试，主要使用理论上可能的尽可能长的代码。发现脉冲串长度为2、3和4的代码。多项式的次数从6到18不等，给出了宽范围的码长。在单相邻码和双相邻码的情况下，给出了存在最大理论码长的一个充分条件。本文详细研究了长度为2^p + 1的码的所有单错和双错的纠正问题。作为三重纠错的一种特殊情况，本文还对长度为23的Golay码进行了分析。最后，这些和其他已知的移位寄存器代码的例子进行了比较，一个穷举搜索的结果，为合适的多项式的次数为8，9，10和11。
A class of shift-register codes is studied for correction of multiple errors. The defining matrix will satisfy an irreducible polynomial equation with coefficients 0 or 1. The errors are classified in terms of error cycles and a simple procedure is suggested to determine if error cycles are distinguishable. To facilitate the analyses, another classification of errors is introduced leaving the number of erroneous digits (weight) invariant. A number of specific codes have been investigated either for correction of error bursts or correction of all multiple errors of certain weights. Burst lengths 2, 3, 4, 5 and 6 are tried, mainly with as long codes as are theoretically possible. Codes are found for burst lengths 2, 3 and 4. The degree of the polynomial varies from 6 to 18 giving a wide range of code lengths. With single and double adjacent errors, a sufficient condition is given for the existence of a code of maximal theoretical length. Correction of all single and double errors in a code of length 2^p + 1 is studied in some detail. As a special case of triple error correction, the Golay code of length 23 also is analyzed. Finally these and other known examples of shift-register codes are compared with the result of an exhaustive search for suitable polynomials of degrees 8, 9, 10 and 11.