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BiD Codes: Algebraic Codes from $3 \times 3$ Kernel

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Pith's one-line read This paper introduces BiD codes, binary linear codes of length $3^m$ from a $3\times3$ kernel, and proves their minimum distance grows at least as $N^{0.543}$ at every fixed rate, exceeding the $N^{0.5}$ growth of Reed-Muller codes.

desk verdict A proven N^0.543 distance exponent for a 3x3 kernel code family is the real deal; the simulations are under-supported and should be cleaned up before publication. read the letter →

arxiv 2507.10068 v1 pith:22QMWVZ6 submitted 2025-07-14 cs.IT math.IT

Anirudh Dash , K. R. Nandakishore , Lakshmi Prasad Natarajan , Prasad Krishnan This is my paper · ORCID
classification cs.ITmath.IT MSC 94B0594B6094B65
keywords BiDcodesBermanReed-Mullerminimumdistance3x3kernelabelianpolarsuccessivecancellationorderedsearch
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

BiD codes are binary abelian codes of length $3^m$ that can be read directly off the Kronecker-power matrix $A_3^{\otimes m}$: keep exactly the rows whose Hamming weights lie in a chosen window. The paper's central claim is that for every fixed rate $R\in(0,1)$ there is a sequence of such codes whose minimum distance grows at least as fast as $N^{0.543}$, asymptotically faster than the $N^{0.5}$ growth of constant-rate Reed-Muller codes. That claim rests on a recursive bound that specializes to the closed-form lower bound $d_{\min}\ge\lceil\max\{4^{r_1}3^{m-r_1-r_2},\,3^{m-r_2}2^{r_1+r_2-m}\}\rceil$, and the recursion is tight for every BiD code of length up to $3^5$ except four parameter pairs. Simulations at length 243 in erasure and Gaussian channels put BiD block error rates close to, and sometimes below, those of Reed-Muller, RM-Polar, and CRC-aided Polar codes under essentially maximum-likelihood decoding. If the asymptotic claim holds, BiD codes form a kernel-constructed algebraic family whose rate-distance trade-off crosses the benchmark set by Reed-Muller codes.

What carries the argument

The carrying object is the row-weight recursion for the block matrices $G_{m,w}$: each weight-$w$ submatrix of $A_3^{\otimes m}$ splits as the vertical stack of $(1,1,1)\otimes G_{m-1,w}$, $(1,1,0)\otimes G_{m-1,w-1}$, and $(1,0,1)\otimes G_{m-1,w-1}$, with repetition and single-parity-check codes as the $w=0$ and $w=m$ endpoints. Combined with the direct-sum decomposition $\mathcal{C}_A(m,W)=\oplus_{w\in W}\mathcal{C}_A(m,\{w\})$, this splits any BiD codeword into three shorter codewords and supports an exhaustive case analysis on the number of nonzero slices. The recursion terminates at Berman and dual-Berman codes, whose minimum distances $2^{r+1}$ and $3^{m-r}$ are taken as base cases, and the closed-form bound of Theorem 3.2 is then evaluated along the rate-converging sequence $r_1,r_2\sim 2m/3$. The asymptotic exponent appears after writing the minimum distance in terms of $N=3^m$ and taking the liminf.

What would settle it

Compute the true minimum distance of $\mathrm{BiD}(5,3,3)$, length 243, by exhaustive search or integer programming; the paper's Table I leaves its distance somewhere between 22 and 36, so finding any codeword of weight below 22 would falsify the closed-form lower bound of Theorem 3.2.

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Extended reading notes

Core claim

A Berman-intersection-dual Berman code $\mathrm{BiD}(m,r_1,r_2)$ is the intersection of a Berman code of order $r_1-1$ and the dual of a Berman code of order $r_2$; in the spectral description of abelian codes it is $\mathcal{C}_A(m,\{r_1,\ldots,r_2\})$, meaning exactly those vectors whose frequency-domain entries vanish outside Hamming-weight levels $r_1$ through $r_2$. The main discovery is that this algebraic intersection has a simple kernel description: its generator matrix is the submatrix of $A_3^{\otimes m}$, with $A_3=[1\,1\,1;\,1\,1\,0;\,1\,0\,1]$, consisting of rows with weights between $2^{r_2}3^{m-r_2}$ and $2^{r_1}3^{m-r_1}$. The recursive distance bound (Theorem 3.1) analyzes the three length-$3^{m-1}$ slices of a codeword and yields the two closed-form lower bounds of Theorem 3.2. Choosing $r_1,r_2$ near $2m/3$ makes the rate converge to any prescribed $R$ and makes the exponent $\log d_{\min}/\log N$ approach at least $\log 6/\log 27>0.543$, exceeding the $0.5$ limiting exponent of constant-rate Reed-Muller codes.

Load-bearing premise

The whole distance proof assumes the previously published minimum distances of Berman codes and their duals are correct; if those numbers were wrong, the recursive base cases, the closed-form bound, and the asymptotic exponent would collapse.

Editorial extensions

If this is right

  • At any fixed rate in $(0,1)$, BiD codes provide a constructive sequence with minimum distance growing at least as $N^{0.543}$, so the claimed improvement over Reed-Muller's $N^{0.5}$ applies across the whole rate range, not just at capacity.
  • The recursive bound yields exact minimum distances for all BiD codes of length up to $3^5$ except four parameter choices, so the construction comes with reliable finite-length distance data for practical blocklengths.
  • Because the codes are defined by row selection from a $3\times3$ kernel, the same encoding transform used for polar codes realizes them, and successive-cancellation-style decoders can be adapted without changing the kernel.
  • Simulations at length 243 place BiD and dynamically frozen dBiD codes at or below the block error rates of Reed-Muller, RM-Polar, and CRC-aided Polar codes at matched rates in erasure and AWGN channels, under essentially maximum-likelihood decoding.
  • The observed weight-48 codeword in $\mathrm{BiD}(5,2,2)$ shows the closed-form bound is tight for at least one practical code, and the four untight cases mark where exact distances remain open.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension would apply the same slice recursion to kernels larger than $3\times3$; the asymptotic exponent would then be governed by the row-weight spectrum of the kernel, suggesting a general family of distance-growth lower bounds indexed by kernel size.
  • The non-tight recursion cases (for example $(m,r_1,r_2)=(5,3,3)$) are concrete places to test whether the closed-form bound is loose or the recursion itself loses information; an exact determination there could guide a tighter theorem.
  • Because the paper's AWGN results use essentially-ML SCOS decoding with high node-visit counts, a natural next experiment is to measure BLER at fixed decoder complexity, which could reverse the practical ordering at low latency.
  • The abelian ideal structure of BiD codes may admit transform-domain decoding, which would address the paper's stated bottleneck of high decoding complexity without changing the distance guarantees.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper introduces Berman-intersection-dual Berman (BiD) codes, abelian codes of length 3^m obtained by intersecting Berman codes with dual Berman codes. It shows that these codes admit a simple generator-matrix description as rows of a Kronecker power of a 3x3 kernel, derives recursive bounds on their minimum distance (Theorem 3.1), closed-form lower bounds (Theorem 3.2), exact minimum distances for two special weight sets (Theorems 3.3 and 3.4), and an asymptotic result (Theorem 3.5): for every fixed rate R in (0,1), there exists a sequence of BiD codes with rate converging to R and liminf log(d_min)/log N >= log 6/log 27 ≈ 0.543, exceeding the N^{0.5} growth of Reed-Muller codes at the same rate. The paper also reports simulations at length 243 in the BI-AWGN and erasure channels, using a successive-cancellation ordered-search (SCOS) decoder with claimed essentially-ML performance, and compares favorably with RM, RM-Polar, and CRC-aided Polar codes.

Significance. The main theoretical contribution is substantial and appears correct. Theorem 3.5 provides a proved family of binary linear codes with a minimum-distance growth exponent strictly larger than 1/2 at every fixed rate, improving on the long-studied Reed-Muller family under a simple kernel-based construction. The proof is a genuine lower-bound argument, not a restatement of an assumption: it uses known Berman-code distances as base cases and a CLT-based rate selection. The recursive distance bounds in Theorem 3.1 are supported by a detailed case analysis in Appendix D, and the closed-form bounds in Theorem 3.2 are checked against numerically computed bounds in the tables. The paper also gives an explicit generator-matrix description and an adaptation of the SCOS decoder. The main weakness is experimental: the claimed essentially-ML performance is not fully substantiated due to missing decoder parameters and supporting curves, but this does not affect the correctness of the asymptotic theorem. If the simulation details are clarified or the claims softened, the paper would be a solid contribution to algebraic coding theory.

minor comments (6)
  1. [Section 5.1, Appendix J] The SCOS decoder parameters λmax and η are not reported, and the empirical lower bound on the ML BLER used to justify the 'essentially-ML' claim is not shown in any figure. Please provide these values and the lower-bound curve, or a link to code/data, so that the simulation results are reproducible. Without this, the abstract's statement about ML decoding is not verifiable.
  2. [Appendix F] The proof of the first lower bound in Theorem 3.2 (the expression 4^{r1} 3^{m-r1-r2}) is only sketched with 'the proof of the other expression follows similar steps.' Since this bound is used to generate Fig. 2 at practical lengths, please include a complete induction or a detailed outline to make the theorem fully self-contained.
  3. [Section 3] In the paragraph before Theorem 3.5, 'N = 2 m' appears to be a typo for 'N = 2^m' when referring to RM codes; please correct it.
  4. [Table I] The entry for BiD(5,2,2) lists d_min as 48–54, but Section 5.1 states that simulations show the exact minimum distance is 48. Please update the table entry to 48, or add a footnote explaining the discrepancy.
  5. [Section 2.2] The paper uses the permuted kernel AAA'_3 in simulations but develops the recursion and distance bounds for AAA_3. Please clarify explicitly that Corollary 2.5 and the distance bounds of Section 3 apply unchanged to AAA'_3, because the row sets of a given weight are identical up to row permutation.
  6. [Abstract] The phrase 'larger distance than RM codes asymptotically' may be read as a claim about exact distances; Theorem 3.5 establishes a lower bound on the growth exponent. Consider rewording to 'provably larger asymptotic distance growth' for precision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.5 is a genuine lower-bound proof whose base cases are published Berman-code distance formulas, not a restatement of the target exponent.

full rationale

The derivation chain from Theorem 3.1 to Theorem 3.5 is self-contained in the appendices and does not reduce to a fitted parameter or a self-citation loop. Theorem 3.1 is proved by an exhaustive case analysis on the three length-3^(m-1) subvectors of a codeword; the bounds D2, D3, D4a, D4b and D4' are all defined from minimum distances of shorter BiD codes, and the recursion terminates in the published Berman minimum distances 2^(r+1) and 3^(m-r) (Theorem 2.1). Although Theorem 2.1 cites the authors' prior paper [9], the same formulas are attributed to the independent references [10] and [11], and the Appendix E tables are consistent with them, so the citation is not load-bearing self-citation. Theorem 3.2 is an induction whose base cases check those same external values and whose induction step propagates the two closed forms through the recursion; it is not an assumption of the desired asymptotic exponent. Appendix I then selects r1 and r2 via a standard CLT argument so the rate converges to any R in (0,1), and the second bound of Theorem 3.2 gives liminf log(dmin)/log N >= log 6 / log 27. The kernel-based construction of Theorem 2.4 is proved directly from the weight recursion in Lemma 2.3. The only under-supported part of the paper is the empirical claim that the SCOS decoder parameters lambda_max and eta were chosen large enough to give essentially-ML BLER, since the values are not reported and no code or data are provided; this affects the simulation comparisons, not the algebraic distance theorems, and is a reproducibility concern rather than a circularity. No fitted value is renamed as a prediction, and no uniqueness or ansatz is imported through a self-citation to force the conclusion.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The mathematical claims rest on the published Berman-code framework and on standard DFT/CLT machinery; no data-fitted constants are used in the distance theorems. The only unspecified tuning parameters are in the SCOS decoder, which affects the empirical BLER claim, not the distance bounds.

free parameters (1)
  • SCOS decoder parameters lambda_max and eta = not reported (set 'sufficiently large' to approach ML)
    Section 4 and Appendix J set them large enough that BLER is close to ML, verified only empirically; quantitative values are not given, so the simulation claim's reproducibility depends on unspecified tuning.
assumptions (4)
  • domain assumption The spectral null-code family CA(m,W) and its direct-sum decomposition CA(m,W)=direct sum over w in W of CA(m,{w}) hold as stated.
    Section 2.1 takes the spectral definition and Eq. (1) from [9, Remark 36]; all later generator and distance results depend on this decomposition.
  • domain assumption Minimum distances of Berman and dual Berman codes are dmin=2^{r+1} and dmin=3^{m-r} (Theorem 2.1, from [9]-[11]).
    Used as termination and base cases in the recursive bounds of Theorem 3.1 and in the induction of Theorem 3.2; if incorrect, the distance claims collapse.
  • standard math The DFT over F4 correctly represents the group algebra F2[Z_3^m] for the spectral description of abelian codes.
    Standard finite-field Fourier transform; the paper invokes it in Section 2.1 without proof.
  • standard math The central limit theorem applies to the binomial distribution used to select r1,r2 in Theorem 3.5.
    Appendix I uses the CLT to show that the chosen code rates converge to any R in (0,1); this is standard.

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Pith. "Pith review of BiD Codes: Algebraic Codes from $3 \times 3$ Kernel." pith.science (2026). https://pith.science/paper/22QMWVZ6

@misc{pith2026250710068,
  author       = {Pith},
  title        = {Pith review of: BiD Codes: Algebraic Codes from $3 \times 3$ Kernel},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/22QMWVZ6}},
  note         = {Machine review of arXiv:2507.10068}
}
abstract

We introduce Berman-intersection-dual Berman (BiD) codes. These are abelian codes of length $3^m$ that can be constructed using Kronecker products of a $3 \times 3$ kernel matrix. BiD codes offer minimum distance close to that of Reed-Muller (RM) codes at practical blocklengths, and larger distance than RM codes asymptotically in the blocklength. Simulations of BiD codes of length $3^5=243$ in the erasure and Gaussian channels show that their block error rates under maximum-likelihood decoding are similar to, and sometimes better, than RM, RM-Polar, and CRC-aided Polar codes.

Figures

Figures reproduced from arXiv: 2507.10068 by the authors.

Figure 1
Figure 1. The recursion tree for bounding the minimum distance of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The log-normalized minimum distance of RM codes (length 256) and BiD codes (length 243, using Theorem 3.2) versus rate. is a generalization of the minimum distance expression for dual Berman codes, and the second extends the expression for Berman codes to BiD codes. Theorem 3.2: The minimum distance of BiD(m, r1, r2) is at least ⌈max {4 r1 × 3 m−r1−r2 , 3 m−r2 × 2 r1+r2−m}⌉. Proof: The proof uses induction in a stra… view at source ↗
Figure 4
Figure 4. Rnorm versus code rate in the BEC without (above) and with (below) dynamic freezing. The (m, r1, r2) parameters of some BiD codes are shown [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: BLER of BiD(5, 3, 4) and dBiD(5, 3, 4) (kernel A′ 3 ) in BEC. of BLER) and the erasure channel (under ML decoding) for BiD codes with length 3 5 = 243. We choose this length for the convenience of comparing with codes of length 2 8 = 256 (which is close to 243) obtaine…
Figure 6
Figure 6. Figure 6: BLER of BiD(5, 2, 2) and dBiD(5, 2, 2) (kernel A3) in BEC. I. Proof of Theorem 3.5. Let Z1, . . . , Zm be independent Bernoulli(2/3) random variables. The rate of BiD(m, r1, r2) is 1 3m Xr2 w=r1  m w  2 w = Xr2 w=r1  m w  2 3 w  1 3 m−w = P " r1 ≤ Xm i=1 Zi ≤ r…
Figure 7
Figure 7. Figure 7: BLER of BiD(5, 2, 3) and dBiD(5, 2, 3) (kernel A3) in BEC. respectively. As in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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