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By N.J.A. Sloane
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The proof i~ omitted. 3) Definition The (unextended ) Golay code ~23 is obtained from ~ by deleting the last coordinate from every codeword. 4) Theorem f'§23 is a [ 23,12,7] code, which can be made cyclic by a suitable permutation of the coordinates. 10 that f'§23 is a perfect code. 4 ), encoding is easy. ) 38 § § 3 The Golay Code We shall give two decoding methods, of which the first is the more practical, the second the more elegant. 1) by moving the left hand column to the right): 1 1 G =H = A ( 112 1 11.
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. 1152-1156. [ 89] Weyl, H. J. 1 Introducdon Cyclic codes are the most studied of all codes. After giving some of the general theory in the first few sections, we proceed with a brief description of the Hamming, BCH, Reed-Solomon, and Justesen codes, with a digression on perfect codes.
Let us see how to decode one coordinate, say the first. 16 ). If there is exactly one error, the number of parity checks which fail is either 253 or 77, depending on whether or not the first coordinate is in error. 56 = 112 in the two cases. 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.