Linear Reliability Channel (LRC)

• LRC is a binary-input, binary-output soft-decision channel model that uses rank-ordered reliabilities to define logistic weight rather than traditional Hamming weight.
• Its framework leverages independent but non-identical BSCs and the ORBGRAND strategy to enable closed-form noise laws, capacity, and error-exponent analysis.
• The model motivates a new code-design approach that optimizes for logistic weight structure, promising improved performance in high-noise regimes over conventional methods.

The Linear Reliability Channel (LRC) is a binary-input, binary-output discrete soft-decision channel in which the soft information is the rank ordering of the received symbol reliabilities rather than a real-valued LLR vector. In the formulation introduced in “The Linear Reliability Channel” (Mariona et al., 9 Sep 2025), a block of length $n$ is transmitted together with a uniformly random permutation $T\in S_n$; conditioned on $T$, the ordered positions pass through independent but non-identically distributed BSCs with crossover probabilities

$$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$

so the reliability magnitudes increase linearly with the order index. The model is intended as a tractable approximation to discrete modulation over continuous additive-noise channels in the high-noise regime, and its central analytical consequence is that soft-decision geometry is governed by logistic weight rather than Hamming weight (Mariona et al., 9 Sep 2025).

1. Channel model and observables

The LRC has input alphabet $\mathcal X=\{0,1\}$, hard output alphabet $\mathcal Y=\{0,1\}$, blocklength $n$, noise parameter $\beta\in(0,\infty)$, and an auxiliary soft output $T\in S_n$, where $S_n$ is the set of permutations of $T\in S_n$0. On each block, a permutation $T\in S_n$1 is drawn uniformly at random; conditioned on $T\in S_n$2, the $T\in S_n$3-th position in the reliability ordering is transmitted through a BSC with crossover $T\in S_n$4, and the permutation is then undone to obtain the physical output $T\in S_n$5. Equivalently, for $T\in S_n$6,

$T\in S_n$7

The magnitude of the bitwise LLR is

$T\in S_n$8

so the reliabilities are exactly linear in their rank (Mariona et al., 9 Sep 2025).

The model separates two decoding regimes. In the soft-decision setting, the decoder observes both $T\in S_n$9 and the realized permutation $T$0, so it knows which physical coordinates are rank-1, rank-2, and so on in reliability. In the hard-decision setting, the decoder observes only $T$1 and averages over the uniform prior on $T$2. This distinction is structural rather than cosmetic: conditioning on $T$3 yields independent but non-identically distributed bit flips, whereas marginalizing over $T$4 restores symmetry across coordinates but induces dependence in the effective hard-decision noise process (Mariona et al., 9 Sep 2025).

The noise effect is expressed additively over $T$5. In the soft-decision case one writes $T$6 conditioned on $T$7; in the hard-decision case the corresponding averaged noise is denoted $T$8. The two induced noise laws are the basis for the channel’s dual geometry: soft LRC is controlled by logistic weight, while hard LRC is controlled by Hamming weight (Mariona et al., 9 Sep 2025).

2. High-noise approximation to continuous channels

The LRC is motivated by binary-input continuous-noise channels of the form

$T$9

where the noise density has the location-scale form

$$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$0

with $$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$1 assumed even, strictly log-concave, and $$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$2. This class includes Gaussian, logistic, Laplace, and uniform noise models. The corresponding LLR is

$$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$3

Because $$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$4 is even and strictly log-concave, $$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$5 is strictly increasing, so $$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$6 has a well-defined density $$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$7 (Mariona et al., 9 Sep 2025).

The approximation result is that, as $$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$8, the LLR density is approximately flat near zero: $$q_i=\frac{e^{-\beta i/n}}{1+e^{-\beta i/n}},\qquad i\in[n],$$9 In the high-noise regime, most LLR mass lies near zero, and an almost-flat density implies that the order statistics of $\mathcal X=\{0,1\}$0 are approximately linear in their rank. In consequence,

$\mathcal X=\{0,1\}$1

which is exactly the organizing principle built into the LRC: the sorted reliabilities are treated as linear in the rank index (Mariona et al., 9 Sep 2025).

This approximation is not merely heuristic. It identifies the LRC as a discrete surrogate for continuous-noise soft information in the regime where the ordering of reliabilities carries most of the usable information and their exact magnitudes are secondary. For Gaussian and logistic noise, the approximation improves as $\mathcal X=\{0,1\}$2 grows. For Laplace noise, the source material notes that some reliabilities eventually saturate, so the linear-rank description is accurate only over an initial range of ranks. A plausible implication is that the LRC is most faithful as a low-SNR / high-noise asymptotic model, rather than as a uniform approximation over the full operating range (Mariona et al., 9 Sep 2025).

3. Logistic weight and the combinatorial geometry of the channel

The defining soft-decision statistic of the LRC is the logistic weight. For $\mathcal X=\{0,1\}$3 and $\mathcal X=\{0,1\}$4, it is defined by

$\mathcal X=\{0,1\}$5

Flips in more reliable ordered positions therefore contribute more heavily. Conditioned on the realized permutation, the soft-decision noise PMF depends only on logistic weight: $\mathcal X=\{0,1\}$6 All sequences of the same logistic weight are equiprobable under soft-decision LRC (Mariona et al., 9 Sep 2025).

The hard-decision law has a different structure. If $\mathcal X=\{0,1\}$7 denotes Hamming weight and

$\mathcal X=\{0,1\}$8

then

$\mathcal X=\{0,1\}$9

Thus, after averaging over permutations, all sequences of the same Hamming weight are equiprobable. Hard-decision LRC therefore shares the coarse Hamming partition of the BSC, whereas soft-decision LRC refines it into many more equivalence classes indexed by logistic weight (Mariona et al., 9 Sep 2025).

This refinement is quantified by the logistic coefficients

$\mathcal Y=\{0,1\}$0

These satisfy the symmetry

$\mathcal Y=\{0,1\}$1

and they admit a partition-theoretic interpretation: $\mathcal Y=\{0,1\}$2 equals the number of partitions of $\mathcal Y=\{0,1\}$3 into distinct parts with largest part at most $\mathcal Y=\{0,1\}$4. The normalization identity

$\mathcal Y=\{0,1\}$5

plays the role that the binomial identity plays for the BSC (Mariona et al., 9 Sep 2025).

A further asymptotic description comes from a theorem of Bridges. Writing $\mathcal Y=\{0,1\}$6, defining $\mathcal Y=\{0,1\}$7 implicitly by

$\mathcal Y=\{0,1\}$8

and then

$\mathcal Y=\{0,1\}$9

one has

$n$0

The soft LRC is therefore organized by a combinatorial structure that is finer than Hamming geometry and closer to weighted partition theory. This is the sense in which the paper describes the LRC as having a discrete geometry distinct from that of the BSC (Mariona et al., 9 Sep 2025).

4. Maximum-likelihood decoding and noise guessing

Because the LRC is additive over $n$1, maximum-likelihood decoding can be written as a search over noise patterns. For a code $n$2 and received word $n$3,

$n$4

This places the channel within the GRAND framework: guess noise patterns in decreasing order of probability, subtract each from the received word, and stop at the first resulting codeword (Mariona et al., 9 Sep 2025).

In the soft-decision case, the optimal guessing order is exactly the order of increasing logistic weight. Formally, for fixed $n$5, any guessing function $n$6 satisfying

$n$7

implements ML decoding. The source material identifies this rule with ORBGRAND: it guesses noise patterns in order of ordered reliabilities, which in the LRC coincides with ordering by logistic weight (Mariona et al., 9 Sep 2025).

In the hard-decision case, the optimal guessing order is the order of increasing Hamming weight. Any guessing function $n$8 satisfying

$n$9

is ML for the hard-decision LRC. This is identical to the BSC ordering used by GRAND, although the underlying hard-decision LRC noise law is not the BSC law. The important distinction is that the BSC’s Hamming ordering comes from i.i.d. flips, whereas the LRC’s hard ordering emerges only after averaging over the reliability permutation (Mariona et al., 9 Sep 2025).

The decoding interpretation is therefore exact rather than metaphorical. ORBGRAND is not merely inspired by the LRC; in the LRC soft-decision model it is the ML decoder. Likewise, conventional GRAND is ML for the hard-decision LRC. This yields a rare setting in which hard- and soft-decision GRAND-style decoders admit a direct, closed-form probabilistic analysis under a common channel family (Mariona et al., 9 Sep 2025).

5. Guesswork, capacity, and random-coding error exponents

Let $\beta\in(0,\infty)$0 denote the guesswork of the true noise effect under the optimal guessing function, and define the normalized log-guesswork

$\beta\in(0,\infty)$1

The LRC analysis establishes a large deviation principle for $\beta\in(0,\infty)$2, with rate function obtained from the Legendre–Fenchel transform of the scaled cumulant generating function

$\beta\in(0,\infty)$3

For the soft-decision LRC,

$\beta\in(0,\infty)$4

where

$\beta\in(0,\infty)$5

For the hard-decision LRC,

$\beta\in(0,\infty)$6

with

$\beta\in(0,\infty)$7

and

$\beta\in(0,\infty)$8

The corresponding rate functions $\beta\in(0,\infty)$9 and $T\in S_n$0 are

$T\in S_n$1

with zeros at the Shannon entropy rates and $T\in S_n$2 (Mariona et al., 9 Sep 2025).

These rate functions determine capacity and error exponents for random coding under ML decoding. The soft- and hard-decision capacities are

$T\in S_n$3

For a random codebook of rate $T\in S_n$4, the ML error exponent below capacity is

$T\in S_n$5

where $T\in S_n$6 solves $T\in S_n$7; the same form holds for the hard-decision LRC with $T\in S_n$8 replaced by $T\in S_n$9. The transition point $S_n$0 is the critical rate, below which the exponent is linear in $S_n$1 and above which it becomes strictly convex (Mariona et al., 9 Sep 2025).

A central comparative result is the strict Rényi-entropy ordering

$S_n$2

This implies

$S_n$3

and yields the ordering of error exponents

$S_n$4

It also implies the critical-rate ordering

$S_n$5

Accordingly, soft decision both increases capacity and maintains the linear error-exponent region up to a higher rate. The source material further notes that the soft-decision gain is most pronounced for intermediate $S_n$6: as $S_n$7, hard and soft LRC both approach a BSC with $S_n$8; for very large $S_n$9, both decoding modes become strong and the gap again narrows (Mariona et al., 9 Sep 2025).

6. Code-design implications and terminological context

The LRC suggests a design criterion for soft-decision coding that is different from the classical minimum-Hamming-distance paradigm. Since ML soft decoding is governed by logistic weight, code quality should be assessed with respect to logistic-weight structure or a corresponding minimum logistic distance between codewords, rather than by Hamming distance alone. The source material states that codes designed to be favorable under the LRC geometry may be better suited for soft-decision decoding over AWGN and related channels than codes optimized purely for Hamming distance. It also identifies concrete open directions: constructing code families optimized for logistic weight, extending the framework to non-binary and multi-level settings, and treating channels with memory (Mariona et al., 9 Sep 2025).

The term LRC is, however, heavily overloaded in coding theory. In distributed storage, it usually denotes Locally Recoverable Codes; “Capacity of Locally Recoverable Codes” explicitly states that, in that work, “LRC” always means Locally Recoverable Codes, not “Linear Reliability Channel” (Mazumdar, 2018). A different reliability notion appears in feedback coding for channels with memory, where Burnashev-type bounds take the linear form

$T\in S_n$00

and the source material associates this with a broader “linear reliability” perspective for unifilar channels (Anastasopoulos et al., 2017). In yet another strand, URLLC fading analyses use a log-log linear outage law

$T\in S_n$01

with $T\in S_n$02 interpreted as a slope or diversity-order parameter (Eggers et al., 2017). These usages are related only at the level of mathematical analogy.

The 2025 Linear Reliability Channel is therefore best understood as a specific discrete soft-decision channel model whose soft information is a reliability permutation and whose decoding geometry is logistic rather than Hamming. Its significance lies in making soft-decision ML decoding analytically explicit: the model is discrete, combinatorial, and sufficiently structured to yield closed-form noise laws, exact ML decoding rules for GRAND and ORBGRAND, and random-coding error exponents that quantify the advantage of soft information (Mariona et al., 9 Sep 2025).