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A binary linear code is called {\em LCD} if it intersects its dual trivially.We show that the coefficients of the joint weight enumerator of such a codewith its dual satisfy linear constraints, leading to a new linear programmingbound on the size of an LCD code of given length and minimum distance. Inaddition, we show that this polynomial is, in general, an invariant of a matrixgroup of dimension $4$ and order $12$. Also, we sketch a Gleason formula forthis weight enumerator.

The joint weight enumerator of an LCD code and its dual