By N.J.A. Sloane
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Extra resources for A Short Course on Error Correcting Codes
40 + 5 = 125 in the two cases. To summarize: First Coordinate not in error Number of Errors 1n error 1 253 77 2 176 112 3 141 125 Therefore there is a simple threshold test: if more than 133 out of the 253 parity checks fail, the first coordinate is in error, if less than 133 fail this coordinate is correct. This test can then be applied successively to all the coordinates (using the fact that thc code is invariant under a transitive group ). § 3. 2) For the use of the weight enumerator in obtaining properdes of a code, see for example MacWilliams [ 76], Berlekamp [ 1] Ch.
V) To show that ~ is invariant under T: T sends the first row of G into 0,1,0100011101,1,1,0000000000 which is easily verified tobe the sum of rows 0,2,6, 7,8,10,11, and therefore is in the code. T sends the last row of G into 112 d 2 , which is the complement of the last row, and is in the code by (iv). (vi) We divide each vector of ~ into a left half L and a right half R. Clearly wt(L) =wt(R) = 0 (modulo 2). R will denote the complement of the right half. (vii) <§ contains no vector of weight 4.
1 02-106. J. D. Thesis, Dept. , Harvard University, May, 1962. L. A. Sloane (1972), Generalizations of Gleason's Theorem on Weight Enumerators of Self-Dual Codes, PGIT 18:794-805. L. A. Sloane (1973), An Upper Bound for Self-Dual Codes, IC 22: 188-200. L. A. Sloane (197 ), On the Invariants of a Linear Group of Order 336, Proc. Camb. Phil. , To Appear. F. E. Y. 1961. 58 References [ 88] Molien, T. (1897), Ueber die Invarianten der Linear Substitutionsgruppen, Sitzungsber. Koenig. Preuss. Akad. , pp.