Abstract

A method is established to convert results from finite geometries to error-correcting codes. Using this we can determine the dimension of a maximal self-orthogonal code and the number of self-orthogonal codes of any fixed dimension in a given space. These results are applied to the Golay (11,6) code over GF(3) giving five equivalent conditions for a code to be equivalent to the Golay code. One of these is that every perfect (11,6) code is equivalent to the Golay code. Analogous conditions are established for the codes equivalent to the extended Golay (12,6) code over GF(3). One of these is that every (12,6) code with minimum weight 6 is equivalent to the Golay code. The corresponding theorems for the Golay (23,12) code over GF(2) and the extended Golay (24,12) code over GF(2) are given. Since these results are easier to demonstrate than the results over GF(3), the method of proof here uses known facts about these codes.

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