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《Journal of Wuhan University(Natural Science Edition)》 1980-04
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AN ALGORITHM TO GENERATE THE CHECK MATRIX OF OPTIMAL MINIMUM ODD-WEIGHT-COLUMN SEC- DED CODES

Zhang Huanguo  
A Class of Optimal Minimum Odd-weight-column SEC—DED Codes was presented by M. Y. Hsiao in 1970. This code is now widely used to increase computer memory reliability. In this paper we are going to give an algorithm to generate the check matrix of the code. The check matrix H of the code should satisfy the following three conditions: C1. Every column should have an odd number of 1's, i. e., all column vectors are odd weight, and every two columns are different from each other. C2. The total number of 1's contained in the check matrix should be a minimum. C3. The number of 1's in each row of the check matrix should be made equal, or as close as possible, to the average number, i. e., the total number of 1's in the check matrix divided oy the number of rows. Given R, n, if n=, all the vectors of dimension R with weight (2i+1) should oe used to build the check matrix H. Otherwise, if n≠, there certainly exists another I, such that n. Let D=n-. First we construct submatrix H_1 with all of the vectors of dimension R with weight (2i+1). Then the other D vectors must be selected out of C_R~(2T+1) vectors of dimension R with weight (2I+1) by the following method. Here. we define the cyclic shifting vector one oit as the transformation T in linear space of dimension R over GF(2). Using the transformation T, all C_R~(2I+1) Vectors of dimension R with weight (2I+1) may be classified into some minimum groups, each of such groups can be used to construct a matrix, such that the number of 1's in each row of the matrix is equal to the average number in the matrix. If R|D(2I+1), we construct the submatrix H_2 with corresponding the minimum groups of vector, and let H= [H_1H_2]. If RD(2I+1), but D=D_1+r, where R|D_1(2I+1), 1≤rR. We construct the submatrix H_2 with the minimum groups of vector corresponding to D_1, and H_3 with other r vectors R_G T~(N(2I+1)(m~odR)), N=0, 1…(r-1), which are taken from the vector group{R_G, R_GT…, R_GT~(R-1)} that is generated oy transforming vector R_G=[…100…0]'. let H=[H_1H_2H_3]. In this case the numbers of 1's in every row of the matrix H are not equal, out the maximum differeuce d_(max)=1.
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