On the determination of GRAND noise sequences by employing integer compositions

Abstract

Guessing Random Additive Noise Decoding (GRAND) has been proposed as an efficient and universal decoding technique for linear block codes [1]. A necessary element of the GRAND decoder is a set of noise sequences of length $n$ equal to the code length. The decoded codeword is taken as the first valid codeword that results from the binary addition of the noise sequences, in order of likelihood, with the received word. For a binary code of length $\boldsymbol{n}$ there are $2^{n}$ such noise sequences. When $\boldsymbol{n}$ is large, it will typically be intractable to add every noise sequence during decoding. Moreover, these sequences exhibit a number of error bursts $(\boldsymbol{m})$ and a total number of bit flips $(\boldsymbol{l})$. In [1] the authors provide expressions to determine the probability of each error burst sequence for a two-state Markov channel model. The resulting decoder is called GRAND-Markov Order (GRAND-MO). The authors of [1] describe a GRAND-MO error pattern generator for the two-state Markov channel based upon sequential transition between $(\boldsymbol{m},\boldsymbol{l})$ pairs, incrementing $\boldsymbol{m}$ first, and then treating each value of l. In this paper we address a method of constructing the n-length noise sequences based upon integer compositions on the required $l$ bit flips as well as the $\boldsymbol{n}-\boldsymbol{l}$ non-errors in the sequence. This method allows one to construct noise sequences in order of probability without the need to construct less likely noise sequences. A GRAND decoder may employ a limited number of noise sequences and abandon decoding upon finding a valid code word, or upon reaching an abandonment threshold without decoding success. In this way the most likely noise sequences can be prioritized in decoding.