SGRAND: Soft-Input GRAND Decoding

• SGRAND is a soft-input variant of GRAND that decodes by guessing noise patterns ordered by soft reliability, enabling ML-like performance.
• It achieves near-optimal error rates for short, high-rate block codes while significantly reducing codebook queries through reliability masking.
• Variants like ORBGRAND and DSGRAND balance ML optimality with hardware efficiency, illustrating trade-offs in soft-detection based decoding.

Searching arXiv for recent and foundational papers on SGRAND. Searching arXiv for "Soft GRAND" and related GRAND variants. Soft GRAND (SGRAND) is a soft-input variant of Guessing Random Additive Noise Decoding (GRAND) in which decoding is performed by guessing the additive noise effect, rather than the transmitted codeword, and ordering those guesses with soft detection information. In the GRAND literature, SGRAND denotes both an early simple soft-detection variant that uses a reliability mask and the later full soft-input maximum-likelihood (ML) decoder that uses real-valued reliabilities to rank candidate error patterns. Across these formulations, the central objective is the same: to retain the code-agnostic, noise-centric character of GRAND while exploiting channel reliability information to approach or realize ML decoding for arbitrary block codes, especially in the short-length, high-rate regime (Duffy et al., 2019, Solomon et al., 2020).

1. Origins and conceptual position

GRAND was introduced as a universal ML decoder for block codes that identifies the noise corrupting a codeword instead of directly searching the codebook. SGRAND extends that paradigm to soft detection. In the 5G NR CA-Polar setting, a simple soft detection variant was already described in 2019: the receiver supplies the demodulated vector together with a reliability mask $s^n$, and noise guesses are restricted to positions marked unreliable. In that formulation, bits marked reliable are not modified, which reduces the effective noise space and can lower the number of codebook queries substantially (Duffy et al., 2019).

The 2020 formulation of SGRAND made the stronger claim: the decoder can fully avail of soft detection information and thereby realize soft-input ML decoding for arbitrary high-rate, short-length block codes. This positioned SGRAND as a codebook-agnostic alternative to code-specific soft decoders such as CRC-aided successive cancellation list decoding for CA-Polar codes (Solomon et al., 2020).

A closely related precursor is SRGRAND, which employs binarized symbol soft information. SRGRAND restricts guessing to symbols declared unreliable and performs conditional ML decoding given the reliability mask. For random codebooks, that work derives the capacity of the symbol reliability channel as

$C_{SR} = 1 - \mu H,$

where $\mu$ is the average fraction of unreliable symbols and $H$ is the noise entropy per unreliable symbol (Duffy et al., 2019).

2. Maximum-likelihood decoding by ordered noise guessing

The defining operation in SGRAND is the ordering of candidate error patterns by their likelihood under the observed soft information. In one common formulation, if $\ell_i$ denotes the reliability associated with bit $i$, then a candidate error pattern $e^n$ is scored by

$S(e^n) = \sum_{i=1}^{n} e_i \ell_i.$

The decoded codeword is then recovered as

$$\hat{c}^n = \tilde{c}^n \oplus \arg\min_{e^n:\,\tilde{c}^n\oplus e^n\in\mathcal{C}} \sum_{i=1}^n e_i\,\ell_i,$$

so the first codeword hit under this ordering is the ML decision (Yuan et al., 2022).

Equivalent descriptions appear elsewhere in the literature. In joint detection-decoding form, SGRAND takes demapped bits together with a reliability vector $\mathbf{\Lambda}$, and candidate noise sequences $C_{SR} = 1 - \mu H,$0 are ranked by

$C_{SR} = 1 - \mu H,$1

with lower weights corresponding to higher-likelihood noise sequences. The decoder queries the all-zero noise sequence first and then proceeds in non-increasing order of posterior likelihood (Sarieddeen et al., 2022).

For linear codes, codebook membership is typically verified by a parity-check or syndrome test. In SGRANDAB, the abandonment version, decoding stops either at the first valid codeword or after a fixed number of queries, in which case the decoder returns failure or erasure. The search is implemented as a best-first search over error vectors, commonly using a max-heap or priority queue, and the reported complexity for a maximum of $C_{SR} = 1 - \mu H,$2 queries is $C_{SR} = 1 - \mu H,$3, where $C_{SR} = 1 - \mu H,$4 is the cost of the syndrome check (Solomon et al., 2020).

3. Forms of soft information and the main SGRAND family variants

The literature surrounding SGRAND differs less in its codeword test than in the granularity and representation of soft information. Some variants use a binary reliability mask, some use exact real-valued reliabilities, and others quantize or rank those reliabilities to ease implementation. The common feature is that decoding remains noise-centric.

Variant	Soft input	Ordering basis
SGRAND	Real-valued reliabilities or LLRs	Increasing $C_{SR} = 1 - \mu H,$5
Early mask-SGRAND / SRGRAND	Reliability mask $C_{SR} = 1 - \mu H,$6 or 1-bit symbol mask	Guess only on unreliable positions
ORBGRAND	Rank order of reliabilities	Increasing sum of ranks / logistic weight
DSGRAND	Quantized reliabilities	Increasing sum of quantized weights

In the early mask-based formulation for BPSK in AWGN, the mask is built by thresholding the received amplitude: values outside $C_{SR} = 1 - \mu H,$7 are declared reliable and values inside are declared unreliable. The threshold is chosen so that the probability of any bit being incorrectly marked reliable remains acceptably low, with

$C_{SR} = 1 - \mu H,$8

This scheme trades full soft precision for a sharply reduced search space (Duffy et al., 2019).

ORBGRAND replaces exact reliabilities with their rank order. That substitution makes the decoder suitable for efficient implementation, but the literature repeatedly contrasts it with SGRAND: SGRAND is ML-optimal because it orders test error patterns according to the full soft input, whereas ORBGRAND schedules error patterns solely according to reliability ranking and can therefore be suboptimal (Abbas et al., 2021, Wan et al., 26 Jun 2026).

DSGRAND generalizes this idea to conventional quantization. In that setting, real-valued reliabilities are mapped to $C_{SR} = 1 - \mu H,$9 categories, and candidate patterns are tested in increasing sum of quantized weights. For a $\mu$0 CRC-concatenated polar code, DSGRAND with $\mu$1 comes within $\mu$2 of ML decoding and outperforms CA-SCL decoding with a list size of 128 codewords; the basic ORBGRAND matches or outperforms CA-SCL with list size 64 and 3 bits of quantized soft information. Both ORBGRAND and DSGRAND exhibit approximately an order of magnitude less average complexity and two orders of magnitude smaller memory requirements than CA-SCL (Yuan et al., 2022).

A persistent point of comparison is therefore not whether the decoder guesses noise—that is common to all GRAND variants—but how much of the soft observation is preserved in the ordering rule. Within that spectrum, SGRAND occupies the fully soft, ML-ordered end.

4. Error-rate performance, query complexity, and implementation tension

Published results consistently place SGRAND at or near the ML benchmark for short, high-rate codes. For the 5G NR CA-Polar $\mu$3 code, SGRANDAB achieves a block error rate of $\mu$4 at $\mu$5 with mask error rate $\mu$6, whereas hard-decision GRANDAB requires about $\mu$7 for the same BLER. Under the same conditions, SGRANDAB averages fewer than 12 codebook queries per block, while the hard-decision method needs about 110 queries (Duffy et al., 2019).

In the full soft-input formulation, SGRANDAB on a $\mu$8 CA-Polar code with BPSK outperforms CA-SCL by about $\mu$9 at $H$0, even when CA-SCL uses list size $H$1. For codes of length 64 with QPSK, the reported improvement reaches about $H$2 (Solomon et al., 2020).

The principal implementation difficulty is equally clear in the literature. SGRAND’s ordering depends on the received soft values for each block, so the test-error-pattern schedule is data-dependent and inherently sequential. On that basis, it is described as not suitable for parallel hardware implementation, whereas ORBGRAND is suitable for parallel hardware implementation but has lower decoding performance than SGRAND (Abbas et al., 2021).

Later work has tried to resolve that tension without abandoning ML ordering. A parallel design based on a unified binary EP tree preserves ML optimality while enabling batch exploration, pruning, and recursive computation. Reported numerical experiments give a $H$3 acceleration for parallel SGRAND relative to serial implementation, while a hybrid enhanced method achieves $H$4 acceleration (Wan et al., 2 Oct 2025). In a different direction, List-GRAND uses the hardware-friendly candidate generation of ORBGRAND but selects among a list of valid codewords by likelihood, enhancing ORBGRAND’s decoding performance by $H$5–$H$6 at a target FER of $H$7 and incurring a $H$8 area overhead in VLSI (Abbas et al., 2021).

5. Soft output, iterative decoding, and joint detection

Although SGRAND is naturally a soft-input hard-output decoder, later work extended soft-input GRAND algorithms to produce soft output. In joint detection and decoding, GRAND can generate bit LLRs through successive computations of Euclidean-distance metrics corresponding to candidate noise-recovered words. For bit $H$9 in symbol $\ell_i$0,

$\ell_i$1

and when the decoder only explores a shortlist of codewords, approximate LLRs are formed from the difference between the minimum Euclidean distance among visited candidates and the minimum distance among candidates with the complementary bit value. LLR saturation is applied when no complementary codeword is found within the guess budget (Sarieddeen et al., 2022).

That same work introduced turbo-GRAND, in which the generated LLRs at one iteration serve as enhanced a priori information for the next iteration. Simulations show that a few turbo-GRAND iterations match the performance of ML-detection-based soft-GRAND in both AWGN and Rayleigh fading channels, while the average complexity grows linearly rather than exponentially with the number of symbols. In Rayleigh fading, turbo iterations provide up to $\ell_i$2 SNR advantage at $\ell_i$3 compared with ZF-based soft detection (Sarieddeen et al., 2022).

Soft-output GRAND was then formalized at the block level. Any soft-input GRAND algorithm can provide an a posteriori estimate of the likelihood that a decoding is correct, or that the correct codeword is in a list, by accumulating the probabilities of already queried noise effects during decoding. The reported implementation overhead is negligible in computation and memory, and the resulting reliability output can be thresholded to trade undetected errors against block errors (Galligan et al., 2023). Subsequent work refined these estimates by exploiting linear code constraints, especially for even codes, and evaluated accuracy with the Brier Score; the resulting soft output is reported to be essentially as accurate as the maximum a posteriori estimate (Feng et al., 20 Mar 2025).

This progression turns SGRAND from a one-shot ML decoder into a component of SISO and iterative architectures. A plausible implication is that GRAND’s noise-centric search is compatible not only with hard decisions or single codeword output, but also with probabilistic reliability interfaces that are usually associated with list or message-passing decoders.

6. Symbol-level, fading, and channel-memory generalizations

The SGRAND research line has broadened from bitwise additive-memoryless settings to symbol-level processing, fading channels, channel-estimation uncertainty, and channels with memory. A notable development is symbol-level soft decoding without soft demapping. In that approach, ORBGRAND operates directly on symbol-level likelihoods by examining the $\ell_i$4 nearest constellation points to each received symbol, defining the exceedance distance

$\ell_i$5

and ranking candidate symbol sequences by $\ell_i$6. For CA-Polar, CRC, RLC, and BCH codes with 256-QAM, using as few as 3 or 4 nearest neighbors per symbol yields identical block-error rates to bit-level SGRAND and dedicated soft-decision decoders. For a $\ell_i$7 code with 256-QAM, the number of reliabilities falls from 1024 under hard demapping to 256 under symbol-level ORBGRAND with $\ell_i$8 (An et al., 2022).

In fading channels, pseudo-soft GRAND uses colored noise statistics after ZF or MMSE equalization as reliability surrogates rather than full per-bit LLRs. For a CA-Polar $\ell_i$9 code under Rayleigh fading and BPSK, pseudo-soft ORBGRAND is reported to lie within $i$0 of soft CA-SCL with list size 16 and to provide up to $i$1 SNR gain over hard GRAND at $i$2. For BCH $i$3, the corresponding gain over hard GRAND is about $i$4 at the same BLER (Sarieddeen et al., 2022).

When channel estimation errors are significant, GRAND-based decoding has also been paired with multiple channel-estimate candidates. For $i$5 5G NR CA-Polar and CRC codes, using 5 channel candidates gives gains of over $i$6 in BLER relative to decoding that ignores channel estimation error, and integrating residual channel estimation uncertainty into the LLR computation provides an additional $i$7 gain (Wiame et al., 17 Jun 2025).

For channels with memory, the SGRAND principle has been generalized through error bursts and sequence reliability. In linear Gaussian intersymbol interference channels, SGRAND-ISI orders error patterns by total sequence reliability and is proved equivalent to ML decoding. Hardware-oriented approximations such as CDF-ORBGRAND-ISI attain performance within $i$8–$i$9 of the ML lower bound and outperform ORBGRAND-AI by $e^n$0–$e^n$1 at $e^n$2 while using substantially lower computational complexity (Li et al., 9 Mar 2026).

Taken together, these developments show that SGRAND is best understood not as a single fixed decoder but as the exact-soft, ML-ordered pole of a broader GRAND family. That family spans mask-based, rank-based, quantized, soft-output, iterative, symbol-level, and memory-aware decoders, while preserving the original noise-centric decoding viewpoint.