Guessing random additive noise decoding with symbol reliability information (SRGRAND)

TL;DR

This work introduces a scheme for employing binarized symbol soft information within Guessing Random Additive Noise Decoding, a universal hard detection decoder, and incorporates codebook-independent quantization of soft information to indicate demodulated symbols to be reliable or unreliable.

Abstract

The design and implementation of error correcting codes has long been informed by two fundamental results: Shannon's 1948 capacity theorem, which established that long codes use noisy channels most efficiently; and Berlekamp, McEliece, and Van Tilborg's 1978 theorem on the NP-hardness of decoding linear codes. These results shifted focus away from creating code-independent decoders, but recent low-latency communication applications necessitate relatively short codes, providing motivation to reconsider the development of universal decoders. We introduce a scheme for employing binarized symbol soft information within Guessing Random Additive Noise Decoding, a universal hard detection decoder. We incorporate codebook-independent quantization of soft information to indicate demodulated symbols to be reliable or unreliable. We introduce two decoding algorithms: one identifies a conditional Maximum Likelihood (ML) decoding; the other either reports a conditional ML decoding or an error. For random codebooks, we present error exponents and asymptotic complexity, and show benefits over hard detection. As empirical illustrations, we compare performance with majority logic decoding of Reed-Muller codes, with Berlekamp-Massey decoding of Bose-Chaudhuri-Hocquenghem codes, with CA-SCL decoding of CA-Polar codes, and establish the performance of Random Linear Codes, which require a universal decoder and offer a broader palette of code sizes and rates than traditional codes.

Figures (7)

• Figure 1: QPSK subject to uncorrelated bivariate AWGN. Each pair of bits is coded into one of four symbols, indicated by the red dots. (a) displays a heat map of the probability density that the received signal is at a given location. When hard detection is employed, received signals are demodulated to the symbol in the quadrant where the received signal is measured and that is provided to the decoder. (b) shows the minimum Likelihood Ratio (LR) between each hard detection symbol and all others as heat maps, providing a measure of confidence in the hard demodulation. (c) displays a mask of the LR surfaces in (b) where within the hatched area the LR is greater than a threshold, and in the yellow mask area it is less than the threshold, identifying the potentially noise-impacted symbols. In the symbol reliability quantization, symbols received in the yellow masked region are demodulated but are also marked as being potentially noise-impacted. (d) provides heat map views of the probability density function of a received signal, conditioned on it being observed in the yellow masked area of uncertainty.
• Figure 2: Probabilistic guesswork decoding race in a SR-BSC. With $p=0.05$ and codebook rate $R=0.6$, the large deviations rate function for: incorrectly identifying a non-transmitted element of the codebook, $I^U (x)$; guessing the true noise if $q=1$ and all bits are potentially noise-impacted, $I^{N} (x)$; with $q=0.4$, guessing the true noise if a random set of locations are potentially noise-impacted, $I^{{N}^L} (x)$. With $x$ being the value on the x-axis, when $2^{nx}$ noise guesses are made the likelihood of success for each of these three racing elements is approximately $2^{-n\inf_{y<x}I(y)}$ for the relevant rate function, $I(y)$.
• Figure 3: Block error exponent comparison between BSCs with and without symbol reliability information. $pq$, the overall bit-flip probability, is constant. Error exponents are plotted as a function of codebook rate $R$. In the left hand side plot $pq=0.05$, and on the right-hand side $pq=0.01$. Circles indicate Gallager's critical rate. The lowest line has $q=1$ and is the error exponent of the hard detection channel. Higher lines correspond to different $(q,p)$ combinations and have larger error exponents, meaning decoding errors are less likely.
• Figure 4: Approximate block error probability and complexity for BSCs with and without symbol reliability information. The BSC without symbol reliability corresponds to the SR-BSC with $q=1$, and the overall bit-flip probability, $pq$, is constant. (a-b) Show results for $n=100$, $pq=10^{-2}$ and a target block error of $10^{-2}$. In (a), the horizontal dashed line is the target block error and approximate block error probabilities are shown as a function of codebook rate, $R$, for a selection of $(q,p)$ pairs. (b) shows the approximate complexity, in terms of average number of guesses per-bit to identification of a codebook element, which decreases with $q$, even though $p$ is increasing. The dashed black line gives complexity for the approach of BCJR74, Diamonds indicate the rate above which the target block error rate would be exceeded, while the inflection point occurs at cut-off rate. (c-d) show corresponding results for $n=1000$, $pq=10^{-4}$ and a target block error of $10^{-3}$.
• Figure 5: Performance evaluation with RM and RLC $[128,99]$, rate $0.77$, codes in an SR-BSC. Upper panel illustrates GRAND and SRGRAND guessing order on an $n=7$ code, where each column is a putative noise sequence with a dot indicating a $1$, and sequences are queried in order from left to right. Left panel gives BLER performance for majority logic decoding, GRANDAB and SRGRANDAB decoding of a RM code as well as GRANDAB and SRGRANDAB decoding of RLCs. Lower right panel gives average number of codebook queries per bit per decoding for GRANDAB and SRGRANDAB.

Theorems & Definitions (7)

• Proposition 1: Guesswork Moments and Large Deviation Principle Arikan96Pfister04Christiansen13
• Proposition 2: LDP for guessing subordinated noise Christiansen13b
• Proposition 3: LDP for Guessing a Non-transmitted Code-word Duffy18Duffy19
• Theorem 1: Symbol Reliability Channel Coding Theorem
• proof
• Theorem 2: Complexity of SRGRAND and SRGRANDAB
• proof