Certified Short-Cycle Diagnostics for Deployed LDPC Codes: From WiFi to Quantum, with a Covering-Tower Factorization

Abstract

Short cycles in the Tanner graph of a low-density parity-check (LDPC) code degrade belief-propagation (BP) decoding, and counting them is a standard step in code design and evaluation. This paper does not count them faster. It contributes a short-cycle diagnostic whose defining identity is a machine-checked theorem -- the trace-formula gap law tr(A^k) - p_k = tr(B^k), with A the adjacency matrix, B the Hashimoto non-backtracking operator and p_k the power sums of the matching-polynomial roots -- and whose per-code outputs are validated by three mutually independent computations that agree exactly. We report a certified census of the four deployed LDPC codes of the IEEE 802.11n (WiFi) standard at block length n=648, and, to our knowledge the first such diagnostic on a quantum LDPC code, of the IBM "gross code" [[144,12,12]] bivariate-bicycle (BB) code, finding girth 6 and c_6=144 shortest cycles. We then give a covering-tower factorization: a BB code whose Tanner graph is a double cover of a smaller code's Tanner graph inherits its short-cycle profile through the classical Artin-Ihara / 2-lift identity tr B^k(cover) = tr B^k(base) + tr B_s^k(base), verified exactly on the gross/[[72,12,6]] pair (c_6 doubles, 144 = 2*72). The contribution is the theorem-backed, cross-validated, reproducible methodology and its transfer from classical to quantum codes; the underlying mathematics is classical and is not claimed as new. We are explicit about what the certificate does and does not guarantee.