by J.H. van Lint Four new doubly-even (56, 28, 12) codes are constructed from Hadamard matrices of order 28.
We assume that the reader is familiar with the basic facts from the theory of self-dual codes and designs. Our terminology and notation follow [3, lo].
The following theorem provides a method for the construction of doubly-even self-dual codes from Hadamard matrices of order n = 4 (mod 8).
Theorem (Tonchev [ll]). Let H be a Hadamard matrix of order n = 8t + 4 such that the number of +1's in each row and column is congruent to 3(mod 4). Then the following matrix
(6 (H + JY2)
(1)
generates a binary self-dual doubly-even code C of length 2n. The minimum distance of C is at least 8 if and only if each row and column of H contain at least seven + 1's.
This theorem gives a simple criterium for extremality of codes arising from Hadamard matrices of order 8, 12, and 20. Starting from a particular Hadamard matrix, one can transform it into many different (but equivalent) matrices by multiplying rows and columns by -1 so that z!l rows and columns contain a number of +1's congruent to 3 modulo 4. A computer search showed that at least 79 inequivalent extremal doubly-even (40,20,8) codes arise from the three Hadamard matrices of order 20 [ll].
In this note we summarize the results of a search for extremal doubly-even (%,28,12j codes obtakz;: L-known Ikdamard matrices of order 28. 11 till1 According to the above theorem, a necessary condition for the extremality of a *On leave from the Institute of Mathematics, Bulgarian Academy of Sciences,