Leakage Geometry and Bounded-Read Security for Permuted Codes

Christ, Golowich, Gunn, Moitra, and Wichs~\cite{CGGMW26} formulated the Permuted Codes conjecture, which predicts pseudorandomness of noisy codewords under repeated use of a hidden coordinate permutation and alphabet mask. We study the resulting correlations for binary linear codes. Their Fourier coefficients are exactly determined by zero-sum tuples of dual words and the probability that the span of such a tuple lies in the permuted dual code. This gives a rank hierarchy for leakage, together with exact pairwise and statistical-query spectra. At the first boundary of each rank, the surviving tuples are minimum dual circuits. We bound their aggregate Fourier energy using packing in fixed-composition slices, a telescoping orbit count, and weighted enumeration of coordinate labels. For polynomial dual distance, the bounds hold through a logarithmic rank range and extend to relations of arbitrary sample order with small total weight excess. We also construct BCH-type code families for which the low-degree likelihood-ratio norm is exponentially large just above the first threshold, although the two-sample law is exponentially close to uniform in total variation. On the computational side, we prove that bounded independence followed by independent noise fools polynomial-width oblivious read-$r$ branching programs in any prescribed order. The read count may grow on the order of $\log n/\log\log n$ for codes of polynomial dual distance, strengthening the read-once evidence for the Permuted Codes conjecture. The containment viewpoint also leads to shared-seed constructions for hidden linear maps.