By N.J.A. Sloane
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Additional resources for A Short Course on Error Correcting Codes
4. 10) is a special case of a theorem of Perron [ 78] . 14) has been studied by Assmus and Mattsou ,  , Goethals , and many others. Pless [ 79] has shown that the Golay code is unique. 17 are taken from Conway. 16) is from Goethals [ 6 7] . 6) Berlekamp [ 62 ] also considers which error patterns of weight 4 can be corrected. F. , (1966), Perfeet Codes and The Mathieu Groups, Arch. Math. 17:121-135. F. , (1969), New 5-Designs, JCT 6:122-151. R. (1971), Coding Theory and the Mathieu Groups, IC 18:40-64.
4) The code ~ was first discovered by Golay [ 25] , [ 26] . 1) was found by Paige [77 ], and later rediscovered by Leech  and Karlin [ 69]. The discussion given here follows Karlin and MacWilliams [ 70 ] - see this reference for a proof of Theorem (3. 4). Golay gave another perfect code in [ 25], an [ 11,6,5] code over the field GF(3). It can be defined by the generator matrix G = [ 16 lll;ll J where A is a circulant with first row (0,1,-1,-1,1) (van Lint , p. 103). 1 0. In 1967 Leech [ 72 ] used the Golay code to construct a very dense sphere packing in 24-dimensional Euclidean space.
I yi Finding the weight enumerator of a large code is m general very difficult. 3 Extending. 1) Lemma Let c be any (n,M,d) code with d odd. By adding a 0 at the end of each codeward of even weight, and a 1 at the end of each codeward of odd weight, we obtain a new (n + 1,M,d +1) code c'. Extending a Code 31 Proof. 3 ), the distance between any two vectors of c' is even. The minimum distance cannot be less than d (which is odd), so must be d + 1. D. 5) we obtain the (8,16,4) extended Hamming code.
A Short Course on Error Correcting Codes by N.J.A. Sloane