Closed-form characterization of the parity-code weight enumerator

Derive a concise closed-form expression for the logarithm of the weight enumerator value \(\Phi(\gamma n)\), or equivalently for \(n^{-1}\log_2\Phi(\gamma n)\), where \(\Phi\) is the weight enumerator of the blockwise parity code \(\widetilde{P}_{\rep}=P_{\rep}^{n/\rep}\).

Background

The distance analysis introduces $\Phi(c)$ as the number of weight-$c$ codewords in the outer blockwise parity code $\widetilde{P}_{\rep}$. The spectral-shape recursion would ideally begin with the exact normalized logarithm of this enumerator, namely $n^{-1}\log_2\Phi(\gamma n)$.

Because no convenient closed form is available, the paper instead defines $\varphi(\gamma)$ as a deliberately chosen upper bound on this quantity. The resulting spectral-shape functions and subsequent distance estimates therefore rely on bounds rather than an exact characterization of the outer weight enumerator.