Generalized Spectra of Binary BCH Codes

Generalized weight spectra of binary linear codes are studied. An analog of the MacWilliams identities relating generalized spectra of a linear binary code and its dual code is established. Asymptotic ($n\to\infty$) expressions for elements of the $r$-spectrum of a $t$-error-correcting binary BCH code of length $n=2^m-1$ are obtained under some restrictions on $t$. This spectrum is computed in the particular case where $r=2$, $t=2$ and $m$ is odd.