Capacity-achieving Guessing Random Additive Noise Decoding (GRAND)

TL;DR

This paper introduces GRAND, a novel decoding algorithm that achieves capacity in discrete channels by ranking noise sequences, and its approximate variant GRANDAB, which balances complexity and performance.

Contribution

The paper presents GRAND, a capacity-achieving ML decoding algorithm based on noise sequence ranking, and introduces GRANDAB, an efficient approximation with proven capacity achievement.

Findings

GRAND is capacity-achieving with random code-books.

GRANDAB balances decoding complexity and capacity achievement.

Decoding schemes outperform brute-force approaches in Markovian noise examples.

Abstract

We introduce a new algorithm for realizing Maximum Likelihood (ML) decoding in discrete channels with or without memory. In it, the receiver rank orders noise sequences from most likely to least likely. Subtracting noise from the received signal in that order, the first instance that results in a member of the code-book is the ML decoding. We name this algorithm GRAND for Guessing Random Additive Noise Decoding. We establish that GRAND is capacity-achieving when used with random code-books. For rates below capacity we identify error exponents, and for rates beyond capacity we identify success exponents. We determine the scheme's complexity in terms of the number of computations the receiver performs. For rates beyond capacity, this reveals thresholds for the number of guesses by which if a member of the code-book is identified it is likely to be the transmitted code-word. We introduce an approximate ML decoding scheme where the receiver abandons the search after a fixed number of queries, an approach we dub GRANDAB, for GRAND with ABandonment. While not an ML decoder, we establish that the algorithm GRANDAB is also capacity-achieving for an appropriate choice of abandonment threshold, and characterize its complexity, error and success exponents. Worked examples are presented for Markovian noise that indicate these decoding schemes substantially out-perform the brute force decoding approach.