On ternary self-dual codes of length 24

A hstract-A partial classification is given of the self-dual codes of length 24 over GF(3). The main results are as follows: there are exactly two codes with minimum Hamming distance d = 9; most of the codes have d = 6 and are indecomposable; one code with d = 6 has a trivial automor-phism group (this is the first such self-dual code that has been found); the codes generated by the 59 inequivalent 24 X 24 Hadamard matrices have been investigated and there appear to be only nine inequivalent codes (two with d = 9 and seven with d = 6); and in all there are 27 decomposable codes, at least % indecomposable codes with d = 6, and the total number of inequivalent codes is at least 140. T HE TERNARY self-dual codes with a length less than or equal to 20 have been completely classified in [4], [8], and [ 111. This paper gives a partial classification of the ternary self-dual codes of length 24. The following are the main results. 1) There are exactly 27 decomposable codes (see Section II). 2) Let the numbers q be defined by where the sum is over all indecomposable inequivalent codes C containing exactly 2i codewords of weight 3. Let us say that a code is of ;tvpe ie, +je, if it contains exactly 2i + 8j codewords of weight 3 and these codewords form a code equivalent to ie3@je, (see [ 111). Then q gives the total " mass " of the indecomposable codes of type ie,. These numbers are derived in Section III and are shown in Table I. 3) A computer program has been developed to find the full automorphism group of a code. This program will be described in detail later [6]. 4) There is a close connection between certain ternary self-dual codes of length 24 and the Hadamard matrices of order 24. The latter have recently been classified: there are exactly 59 inequivalent matrices [5]. Any of these matrices generates a self-dual code of length 24 (see Theorem 4). Using the computer program described in 3), we have investigated the codes generated by these 59 matrices and their transposes: there are exactly two codes with minimum