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The MacWilliams identities relate the weight enumerators of a code and its dual code. I treated the version for linear codes in my combinatorics class, but felt unsatisfied because I didn't have a use for them (and judging from the feedback, some students shared this feeling).

So... do you know of any neat application of the MacWilliams relations? Preferably one that I can treat in one lecture. I'm thinking about trying to prove the non-existence of codes with certain parameters, but anything is welcome.

One application that I'm aware of is the role these relations played in the proof of the nonexistence of a projective plane of order 10, but this is not ideal material for a lecture.

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Some applications are mentioned in Elkies' Lattices, Linear Codes, and Invariants, Part II (ams.org/notices/200011/fea-elkies-2.pdf). –  Qiaochu Yuan Feb 2 '12 at 21:09

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There is a very nice proof that there is no projective plane with order 6 mod 8 in Assmus, E. F., Jr.; Maher, David P. Nonexistence proofs for projective designs. Amer. Math. Monthly 85 (1978), no. 2, 110–112. This uses the weight enumerator.

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Thanks, Chris! That's just the sort of thing I was looking for. –  fan Feb 6 '12 at 20:34

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