Cubic tensor product codes — three-dimensional generalizations of product codes with parameters (n3,k3)(n^3,k^3) — were recently shown to be iteratively decodable with soft-output GRAND component decoders, yielding long, low-rate codes that converge in few iterations, but only with a single static extrinsic-weighting factor α≈0.7\alpha\approx 0.7 and a small, ad-hoc selection of component codes. The unresolved problem is to determine the optimized decoding configuration for this construction: whether per-code, per-SNR, and per-iteration-stage adaptive extrinsic weighting improves on the static choice, and which component-code families, lengths, and redundancies yield the best end-to-end block/bit error rate and latency trade-offs. The cubic-tensor extrinsic-weighting optimization and component-code characterization questions posed below remain open.

Background

Forward error correction (FEC) codes for soft-detection channels are commonly built by concatenating short component codes and decoding them iteratively with Soft-Input Soft-Output (SISO) decoders, where each component decoder consumes and produces log-likelihood-ratio (LLR) reliability values. A product code arranges codewords of an (n,k)(n,k) component code in a k×kk\times k information array and encodes all rows then all columns, giving an (n2,k2)(n^2,k^2) code; a cubic tensor code extends the same idea to three dimensions, giving an (n3,k3)(n^3,k^3) code whose rate R3R^3 drops cubically in the component rate RR, producing long, powerful, low-rate codes from short components. Turbo product (block turbo) decoding, introduced by Pyndiah, iterates SISO component decoding over the rows, columns, and (for cubic codes) tubes of the array, passing between iterations only the extrinsic information — the part of each decoder output not derived from the input — scaled by a weighting factor α\alpha that damps unreliable extrinsic LLRs in early iterations. Classical turbo product decoding relied on Pyndiah's approximate soft output, which restricted usable component codes to those with efficient hard decoders such as extended Bose–Chaudhuri–Hocquenghem (eBCH) codes. GRAND (Guessing Random Additive Noise Decoding) instead guesses the most likely noise patterns and queries codebook membership, achieving near-maximum-likelihood decoding of any moderate-redundancy code; its soft-input variant ORBGRAND orders queries by bit reliability, and Soft-Output GRAND (SOGRAND, Yuan et al. 2023) supplies accurate blockwise a posteriori LLRs for any component code with moderate redundancy n−kn-k (roughly n−k≤25n-k\le 25), including cyclic redundancy check (CRC) codes, circumventing Pyndiah's approximation and the reliability factor β\beta. With these soft outputs, square product codes built from CRC and eBCH components were shown to be competitive with the low-density parity-check (LDPC) codes of 5G New Radio while converging in fewer iterations. Motivated by applications needing powerful low-rate codes (challenged environments, continuous-variable quantum key distribution), Khalifeh, Duffy, and Médard (arXiv:2412.14017; CISS 2025) extended this to cubic tensor codes: for the CRC (15,10)(15,10) component code they demonstrated the cubic code [3375,1000][3375,1000], and using Koopman CRC components (15,7)(15,7) and (17,8)(17,8) they demonstrated cubic codes like [3375,343][3375,343] and [4913,512][4913,512] with rates near 0.10.1 achieving a block error rate (BLER) of 10−310^{-3} at signal-to-noise ratios where roughly a quarter of demodulated bits are in error, converging in a small number of half-iterations (one half-iteration decodes all rows, all columns, or all tubes, so a full cubic pass takes 1.5 iterations). Two decisions, however, were deliberately left unoptimized. First, whereas prior work on square product codes found a static α≈0.5\alpha\approx 0.5 adequate, the cubic evaluations used a single static α=0.7\alpha = 0.7, chosen from a coarse sweep on one [4096,1331][4096,1331] eBCH cubic code; the authors state it would be possible to further optimize per cubic code by empirically determining an optimal α\alpha, which may depend on the code, the SNR, and the iteration, but for parsimony a single value was used. Second, the reported evaluations covered only a handful of CRC and eBCH components, while the authors note the design space is large — SOGRAND and soft-output guessing-codeword decoding (SO-GCD) apply to any moderate-redundancy component code and soft-output successive-cancellation-list (SO-SCL) decoding covers a large class of polar-like codes — so which component-code families, component lengths, and redundancy regimes give the best end-to-end block error rate (BLER)/bit error rate (BER) and iteration-count (latency) trade-offs for cubic constructions remains unmapped.

Problem Statement

For cubic tensor product codes with parameters (n3,k3)(n^3,k^3) constructed from moderate-redundancy component codes (CRC codes, eBCH codes, and polar-like codes) and iteratively decoded with accurate soft-output component decoders (SOGRAND, SO-GCD, or SO-SCL) under the source's evaluation conventions (complex additive white Gaussian noise (AWGN) channel, binary phase-shift keying (BPSK) modulation, decoding all rows/columns/tubes counting as one half-iteration, abandonment thresholds on the order of 20–30 iterations), determine the optimized decoding configuration, jointly consisting of: (1) the extrinsic-weighting strategy — establish whether the single static weighting factor α=0.7\alpha = 0.7 used in the source can be improved upon by weighting that adapts per component code, per SNR, and per iteration stage, and quantify any resulting gains or absence of gains in block error rate, bit error rate, and mean iteration count across representative cubic codes — at minimum the four cubic codes evaluated in the source (the [3375,1000][3375,1000] CRC, [3375,343][3375,343] and [4913,512][4913,512] Koopman CRC, and [4096,1331][4096,1331] eBCH constructions); and (2) the component-code design-space map — across component-code families, component lengths, and redundancies within the moderate-redundancy regime n−k≤25n-k\le 25 evaluated in the source (extension beyond that regime, where the soft-output decoders permit it, is optional rather than required), determine which configurations yield the best end-to-end BLER/BER and decoding-latency trade-offs for long low-rate code applications, benchmarked against the static-α\alpha cubic baselines and the square product-code reference points used in the source. The design-space map is established by a documented comparison spanning at least two component-code families across several component lengths and redundancy points, together with an explicit rule for transferring a finding to unmeasured component codes. These two deliverables serve one overarching objective — establishing cubic tensor codes as a practically configured and understood code family. Hardware (chip-level) realization of the decoder is out of scope for this task.

References

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