532 THE BELL SYSTEM TECHMCAL JOURXAL, MARCH 1957 



^."'=ri7^{(:)-(-«'-(fl::iD} ^odd 



For 7 = 0, Eq. (35) works out as 



Gf(0)(,.) ^ (1 + xr + n (l + xy^'"-'\l - xY^'"-"'' 



so that 



(0) 



1 + 71 



^ -\- n{ — iy [{ )> s even 



1 + w \\s / \\s 



'.-=r^{(:)+"(-^)'"-(r(::;i)} ^-^ 



Case V: e = 2, ?i = 90 



The double error correcting codes for n = 2, 5 are covered by Cases 

 II, III, respectively. The discovery that 



1 + 90 + K90)(89) = 2'- 



is due to Golay. We have 



2^2(n - 1, ^) = (2^ - n + 1)^' - (n - 1) 



with roots 



i[w - I ±{n - 1)-] 



Since these roots are not integers when n = 90, there can be no 2-code 

 for n = 90.* H. S. Shapiro has shown (in unpublished work) that the 

 Hamming condition for e = 2 is satisfied only in the cases n = 2, 5, 90, 

 so that the only nontrivial 2-codes are those equivalent to the majoritj^ 

 rule code on 5 places. 



Case VI: e = 3, n = 23 



Golay finds: 



1 + 23 + K23)(22) + (23)(22)(21)/6 = 2" 

 and gi\'es explicitly a 3-code on 23 places of group type. We have 



^■M - 1, .^) = (2^ - n + l)[(2s^ - n + 1)"' - (3/^ - 5)] 



and when n = 23 we verify that the roots are the integers 7, 11, 15. 

 Computations by the author show that for n < 10 the Hamming con- 

 dition for e = 3 holds only when n = 3, 7, 23. 



* This settles a question raised by Golay, who shows that there is no code of 

 group type in this case, but not that there is no code at all. 



