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.

Lattices, Linear Codes, and Invariants, Part II(ams.org/notices/200011/fea-elkies-2.pdf). $\endgroup$ – Qiaochu Yuan Feb 2 '12 at 21:09