By N.J.A. Sloane
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Additional resources for A Short Course on Error Correcting Codes
Nice 1970, Gauthier-Villars, Paris, Vol. 3:211-215.  Jacobson, N. (1964 ), Lectures in Abstract Algebra, Vol. J. K. (1973), Spectra of Linear Codes, Third International Symposium on Information Theory, Tallinn, Estonia, June 1973, Abstracts of Papers, Part II, pp. 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.
U. I i =1 . S1 + bJ. wt(B1 s1 ) ~ 3, = BT ST • , ••• ,e 1 ~ = 1, so for some j, 1 ~j ~ 12, Thus the decoding can be done by computing the weights of the 26 vectors (1 ~ j ~12), B1 S 1 , B1 S1 + bj (1 ~ j ~ 12). Example: S=11100 ... 0, wt(S) = 3, socaseI applies. S1 , S = S1 +bj u 1 + u2 + u 3 , so e = 11100 ••• 0. (3. 6. 2) Decoding Metbad II (Goethals [ 6 7 ] ) This is a clever threshold decoding scheme, using the properties of the designs associated with the code. 4 ), any set of codewords can be used as parity checks for the code.
Since a is a primitive n-th root of unity, all these columns are distinct. Thus they comprise all the distinct nonzero binary vectors of length m. , when m = 3, b = 7, and a is a root of x3 + x + 1: H • ~ : : : : : ~ (5. 8. 1. 3). 4) Theorem The Hamming code is a [ n = 2m - 1 , k = n - m, d code, with r = m parity checks.
A Short Course on Error Correcting Codes by N.J.A. Sloane