Guesswork Analysis of Guessing Decoding of Short Codes

Abstract

This paper addresses: how many guesses are needed for maximum-likelihood (ML) performance by guessing decoding of short binary linear block codes—guessing random additive noise decoding (GRAND) and guessing codeword decoding (GCD)? We show the required guesswork depends weakly on code structure: codes with the same length n and dimension k and comparable ML performance need a similar number of guesses. For random codes, the GRAND guesswork is on the order εRCU2^n-k—, where εRCU is the random-coding union bound, tightening the trivial order 2^n-k. We derive a universal upper bound on the number of (partial) test error patterns (TEPs), which is then evaluated accurately with a saddlepoint approximation. Analysis and simulations show that, in terms of guesswork, GCD never exceeds and often improves on GRAND, notably at low rates. To further reduce guesswork, we introduce an ordered-statistics decoder with local constraints (LC-OSD), derived by extending the most reliable basis and incorporating an early stopping rule. To reduce the decoding latency, we propose a parallel LC-OSD that utilizes pre-stored TEPs. Simulations of binary images of Reed-Solomon codes, which approach the randomcoding union (RCU) bound, confirm our analysis and demonstrate the superiority of soft-weight ordering over Hamming-weight ordering.