Decoding Balanced Linear Codes With Preprocessing
Andrej Bogdanov, Rohit Chatterjee, Yunqi Li, Prashant Nalini Vasudevan
TL;DR
This work addresses decoding corrupted codewords for binary linear codes under a strengthening balance assumption and with preprocessing. It introduces a preprocessing framework that outputs an advice matrix $H$ from a distribution over short dual-codeword sums, enabling a decision procedure based on the Fourier bias $\hat{D}_C(w)$ and a subsequent search that iteratively flips bits to minimize proximity to the code. The main contributions are a polynomial-time decoding-with-preprocessing algorithm achieving error tolerance $O((\log n)^2 / n)$ for $n^{-\Omega(1)}$-balanced codes, a quantitative lower bound showing limitations for polynomial-size threshold distinguishers, and an analysis connecting this approach to information-set decoding, worst-case to average-case reductions, and LPN-type cryptographic considerations. The results imply improved decoding performance for random linear codes under preprocessing and highlight potential vulnerabilities in LPN-based cryptosystems when code parameters are publicly available. The paper thus advances understanding of when preprocessing can meaningfully boost decoding and clarifies the inherent limits of decision-based approaches in this regime.
Abstract
Prange's information set algorithm is a decoding algorithm for arbitrary linear codes. It decodes corrupted codewords of any $\mathbb{F}_2$-linear code $C$ of message length $n$ up to relative error rate $O(\log n / n)$ in $\mathsf{poly}(n)$ time. We show that the error rate can be improved to $O((\log n)^2 / n)$, provided: (1) the decoder has access to a polynomial-length advice string that depends on $C$ only, and (2) $C$ is $n^{-Ω(1)}$-balanced. As a consequence we improve the error tolerance in decoding random linear codes if inefficient preprocessing of the code is allowed. This reveals potential vulnerabilities in cryptographic applications of Learning Noisy Parities with low noise rate. Our main technical result is that the Hamming weight of $Hw$, where $H$ is a random sample of *short dual* codewords, measures the proximity of a word $w$ to the code in the regime of interest. Given such $H$ as advice, our algorithm corrects errors by locally minimizing this measure. We show that for most codes, the error rate tolerated by our decoder is asymptotically optimal among all algorithms whose decision is based on thresholding $Hw$ for an arbitrary polynomial-size advice matrix $H$.