On irredundant orthogonal arrays

Maryam Bajalan; Peter Boyvalenkov

REVIEW 2 major objections 4 minor 29 references

On irredundant orthogonal arrays

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For any linear code, at least one of the code and its dual is an irredundant orthogonal array.

desk verdict Nice short paper with a clean duality observation, but the covering-radius section is wrong as written and needs correction before acceptance. read the letter →

arxiv 2506.03688 v1 pith:S3EYDRB7 submitted 2025-06-04 cs.IT math.IT

Maryam Bajalan , Peter Boyvalenkov This is my paper

classification cs.ITmath.IT MSC 05B30

keywords IrredundantorthogonalarraysLinearcodesDualSelf-dualMDSReed-MullerCoveringradius

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

An irredundant orthogonal array (IrOA) is an orthogonal array in which every projection onto $n-t$ columns has all rows distinct, and the paper's central observation is that this is exactly the condition that the array's minimum Hamming distance is at least $t+1$. For linear codes, that observation turns duality into a simple comparison: a code $C$, viewed as an orthogonal array of strength $d_\perp-1$, is irredundant precisely when its own minimum distance $d$ is at least $d_\perp$. Hence every linear code yields an IrOA in at least one of the two dual directions, and codes with $d=d_\perp$—self-dual codes in particular—yield IrOAs in both directions. The paper constructs concrete IrOA families from Reed-Muller, generalized Reed-Muller, MDS, and generalized Reed-Solomon codes, and derives bounds on minimum distance and covering radius for IrOAs generally.

What carries the argument

The engine is the equivalence of Lemma 3: an $OA(M,n,q,t)$ is irredundant iff its minimum Hamming distance satisfies $d \ge t+1$. The duality half uses the standard fact that a linear $[n,k,d]_q$ code is an OA of strength $t=d_\perp-1$; substituting the equivalence turns '$C$ is an IrOA' into the inequality $d \ge d_\perp$. This reduces irredundancy of either dual to comparing two code parameters. The same equivalence drives the later metric bounds, where the bound $M \le q^{n-t}$ and sphere-volume counting enter.

What would settle it

For the covering-radius claim, one calculation settles it: the ternary Hamming code of length 4 is an IrOA(9,4,3,2) and, being a perfect code, has covering radius 1, whereas the extremal result would give $\rho=t=2$. For the main duality theorem, an exhaustive check of small binary linear codes, comparing $d$ and $d_\perp$ against whether projections onto $n-t$ columns repeat, would confirm or break it.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 3. For a linear $[n,k,d]_q$ code $C$ with Euclidean dual $C^\perp$ of minimum distance $d_\perp$, the code $C$ is an $OA(q^k,n,q,t)$ with $t=d_\perp-1$, and $C^\perp$ is an $OA(q^{n-k},n,q,d-1)$. Since Lemma 3 says a linear OA is irredundant exactly when its minimum distance is at least $t+1$, $C$ is irredundant iff $d \ge d_\perp$ while $C^\perp$ is irredundant iff $d_\perp \ge d$. Therefore at least one of the two dual codes is always an IrOA, and both are when $d=d_\perp$. The paper uses this to certify families: self-dual codes are always IrOAs; Reed-Muller, generalized Reed-Muller, MDS, and generalized Reed-Solomon codes become IrOAs under explicit parameter inequalities; and for an IrOA of maximal size $M=q^{n-t}$, the paper proves that the covering radius is exactly $t$.

Load-bearing premise

The main theorem rests on the equivalence between irredundancy and minimum distance at least $t+1$; the covering-radius bound additionally rests on treating the sphere-covering volume $V_q(t,\rho)$ as living in dimension $t$ rather than in the full length $n$.

Editorial extensions

If this is right

For every linear code over $F_q$, at least one of $C$ and $C^\perp$ is an IrOA, so irredundant orthogonal arrays arise from any dual pair of linear codes.
When $d=d_\perp$, both the code and its dual are IrOAs; self-dual codes are the canonical examples, and any other code with equal distance to its dual behaves the same way.
The constructions give explicit parameters for IrOAs from Reed-Muller, generalized Reed-Muller, MDS, and generalized Reed-Solomon codes, with conditions such as $n \ge 2k$ for MDS codes.
An IrOA of the maximal size $M=q^{n-t}$ has covering radius exactly $t$, according to the paper's extremal result.
For IrOAs with index $\lambda>1$, the minimum distance lies between $t+1$ and $n-t$, with the sharper upper bound $n-t-m+1$ when the index is at least $q^m$ and $n$ is sufficiently large.

Reading between the lines

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

The $d \ge d_\perp$ comparison suggests a cheap way to generate IrOAs: take any linear code from a table and use whichever of the dual pair has the larger minimum distance; the paper demonstrates the idea on named families but does not run a broad census.
The condition $d=d_\perp$, not self-duality itself, is what produces IrOAs in both directions, so any code with its distance equal to its dual's—self-dual or not—would give a double IrOA.
One could test whether the extremal covering-radius conclusion extends to near-maximal sizes, where $M$ is close to $q^{n-t}$ and the sphere-volume estimate may still be informative.
The quantum motivation behind IrOAs suggests that every linear code's winning dual side supplies a $t$-uniform state without redundant information, making code tables a systematic source for such states.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the main observation is correct and useful, and the code-family sections are fine, but the covering-radius section has a load-bearing error and Theorem 12 is false. I'd engage with it after a revision.

The genuinely new piece is Theorem 3: for any linear code, at least one of C or its Euclidean dual is an IrOA. That is a clean organizing principle. It follows in a couple of lines from the Chen-Zhang characterization (Lemma 3) plus the standard OA-code duality, but it deserves to be stated, and it gives a quick way to generate irredundant arrays from well-known code families. The subsequent RM, GRM, MDS, GRS applications are mostly parameter checks, but they are correct and the MDS/self-dual cases (Theorems 4, 7, 8) are tidy. The paper is honest about which ingredients are known.

The soft spot is Section 4.2. The Hamming ball volume is defined as V_q(t,ρ), using the array strength t as if it were the code length n. Inequality (4) needs the volume in H_n^q, i.e., V_q(n,ρ). The consequence is that Theorem 11 is false as stated, and Theorem 12 falls with it: the claimed lower bound ρ ≥ t for M = q^{n−t} does not follow. The concrete counterexample is the ternary [4,2,3] Hamming code (tetracode), an IrOA(9,4,3,2) with covering radius 1, not 2. The upper bound ρ ≤ t in Theorem 12 is fine; the lower-bound half is wrong. This is not a small typo. The abstract advertises covering-radius bounds, so the error is public. The minimum-distance bounds in Section 4.1 are less exciting but solid; the Singleton-type bound in Theorem 10 is a trivial application.

The citation pattern is fine. Reliance on [9], co-authored by one of the authors, is mild and does not carry the main results.

I would not recommend rejecting on the strength of the false theorem alone, because the structural result is correct and the error is localized and fixable. But the paper should not appear with Theorem 12 as written, and the abstract needs rephrasing until the covering-radius claims are corrected. A referee who reads Section 4.2 carefully will catch this.

Bottom line: worth a serious referee, and the likely outcome should be either acceptance after a required revision that fixes Section 4.2 or a clear request for major revision. I'd bring it to the reading group to discuss the duality observation and to double-check the covering-radius fix.

Referee Report

2 major / 4 minor

Summary. The paper studies irredundant orthogonal arrays (IrOAs), arrays whose projection onto n−t columns has distinct rows, using the characterization that an OA is an IrOA iff its minimum Hamming distance is at least t+1. The main results are: Theorem 3 shows a linear code C is an IrOA iff d ≥ d⊥ and C⊥ iff d⊥ ≥ d, so at least one of a code and its dual is always an IrOA; Theorem 4-8 and the corollaries construct IrOAs from self-dual, Reed-Muller, generalized Reed-Muller, MDS, and generalized Reed-Solomon codes; and Section 4 gives bounds on minimum distance and covering radius, culminating in the claim that an IrOA with M = q^{n−t} has covering radius exactly t.

Significance. If correct, Theorem 3 and the explicit code families would be a useful contribution: Theorem 3 gives a simple, broad source of IrOAs, and the constructions from RM/GRM/MDS/self-dual codes are concrete and largely correct. The proofs of Theorems 1-8 are transparent derivations from known results, and the minimum-distance bounds in Section 4.1 are reasonable. However, the covering-radius contribution is not correct: Theorem 11 uses the wrong Hamming ball dimension and Theorem 12 is false, with an explicit counterexample. The advertised claim that the covering radius is exactly t in the extremal case must be withdrawn. The paper's core combinatorial results remain defensible, but the covering-radius section requires major repair.

major comments (2)

[Section 4.2, Eq. (4) and Theorem 11] The volume of a Hamming ball in H_n^q of radius ρ is V_q(n,ρ) = Σ_{i=0}^ρ C(n,i)(q−1)^i, not V_q(t,ρ). The sphere-covering inequality with |C| = λq^t gives λV_q(n,ρ) ≥ q^{n−t} and, for an IrOA with M ≤ q^{n−t}, V_q(n,ρ) ≥ q^t. Replacing n by t makes both inequalities stronger and false; for example, the ternary [4,2,3] Hamming code is an IrOA(9,4,3,2) with ρ = 1, and V_3(2,1) = 3 < 9 = q^t, so the second inequality of Theorem 11 fails. The first asserted inequality also fails for the same code: λV_3(2,1) = 3 < 9 = q^{n−t}.
[Theorem 12 and Abstract] Theorem 12 is false. The ternary [4,2,3] Hamming code (tetracode) is an IrOA(9,4,3,2) with M = q^{n−t} = 9 and covering radius 1, not t = 2. The upper-bound part of the proof, showing ρ ≤ t, is correct, but the lower-bound step invokes the false second inequality of Theorem 11, and no alternative argument is provided. Accordingly, the abstract and Section 1 statements that the covering radius is exactly t in the extremal case M = q^{n−t} are unsupported and must be corrected or removed.

minor comments (4)

[Theorem 7 (proof)] The proof cites Lemma 2 to justify the irredundancy conditions; it should cite Lemma 3, which is the characterization of IrOAs via minimum distance.
[Theorem 6(2) (proof)] In the proof of part (2), the line 'R_q(m,r)^\perp = OA(2^{k'},2^m,2,...)' appears to be a copy-paste from the binary case; the parameters should use q, not 2.
[Throughout Section 4.2 and Theorems 5-6] There are several typos: 'assumtion' (Theorem 6 proof), 'detemine' (before Theorem 12), and 'necessary condition for redundancy' should be 'necessary condition for irredundancy' in Theorems 5 and 6.
[Theorem 2 and Theorem 12] In the extremal case M = q^{n−t}, Theorem 2 already forces n = 2t and λ = 1; stating this explicitly before Theorem 12 would clarify the setting, though it does not affect the counterexample.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central IrOA dual-code theorem and code constructions are direct consequences of standard coding-theory facts; the only overlapping-author citation is peripheral, and the covering-radius flaw is a dimensional correctness error, not a circular reduction.

full rationale

The paper's main derivation chain is self-contained. Theorem 3 follows from the standard fact that a linear code C is an OA of strength d⊥−1 (Hedayat–Sloane–Stufken) together with Lemma 3 (irredundancy iff d≥t+1); no fitted quantity or target conclusion is used as an input. The RM, GRM, MDS and GRS constructions invoke only known code parameters and Theorem 3, so they are not circular. The only citation with overlapping authorship is [9] (Boumova–Boyvalenkov–Stoyanova), used for the conditional bound d≤n−t−1 in Corollary 3(2); that is a published external theorem with stated hypotheses, and the paper's main new minimum-distance estimate (Theorem 10) is proved independently from the Singleton bound. Section 4.2 contains a serious correctness problem: the sphere-covering ball volume is written as V_q(t,ρ) with t (the array strength) instead of the code length n, so Theorems 11 and 12 are not valid as stated (e.g., the ternary [4,2,3] code is an IrOA(9,4,3,2) with covering radius 1). This is a mathematical error in a proof, not a circularity: the paper does not fit parameters or define conclusions into premises, and the main structural results are unaffected.

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

The paper relies on standard coding theory results (Delsarte-type duality, Singleton bound, RM/GRM/MDS dualities) and one imported characterization of irredundant arrays. No new parameters are fit to data, and no new physical or combinatorial entities are invented. The only problematic input is the incorrect dimension in the covering-radius volume, which is a flaw rather than a legitimate axiom.

assumptions (4)

standard math A linear [n,k,d] code C forms an orthogonal array of strength t = d⊥ - 1 (Hedayat, Sloane, Stufken, Theorem 4.6, cited in Section 3.1).
This is the bridge between codes and orthogonal arrays; Theorem 3 depends entirely on it.
standard math An OA is irredundant if and only if its minimum distance is at least t+1 (Chen-Zhang [10, Lemma 2.1], restated as Lemma 3).
This imported characterization is used throughout the paper, including the code constructions and the bounds.
standard math The Singleton bound d ≤ n - k + 1 for q-ary codes, used in Theorem 10.
This bound underlies the minimum-distance upper bounds for IrOAs.
standard math The Delsarte bound ρ ≤ n - t for orthogonal arrays, cited in Section 4.2.
This bound is used to derive the covering radius results, though the paper incorrectly replaces n with t in the volume computation.

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Cite this review

Pith. "Pith review of On irredundant orthogonal arrays." pith.science (2026). https://pith.science/paper/S3EYDRB7

@misc{pith2026250603688,
  author       = {Pith},
  title        = {Pith review of: On irredundant orthogonal arrays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S3EYDRB7}},
  note         = {Machine review of arXiv:2506.03688}
}

abstract

An orthogonal array (OA), denoted by $\text{OA}(M, n, q, t)$, is an $M \times n$ matrix over an alphabet of size $q$ such that every selection of $t$ columns contains each possible $t$-tuple exactly $\lambda=M / q^t$ times. An irredundant orthogonal array (IrOA) is an OA with the additional property that, in any selection of $n - t$ columns, all resulting rows are distinct. IrOAs were first introduced by Goyeneche and \.{Z}yczkowski in 2014 to construct $t$-uniform quantum states without redundant information. Beyond their quantum applications, we focus on IrOAs as a combinatorial and coding theory problem. An OA is an IrOA if and only if its minimum Hamming distance is at least $t + 1$. Using this characterization, we demonstrate that for any linear code, either the code itself or its Euclidean dual forms a linear IrOA, giving a huge source of IrOAs. In the special case of self-dual codes, both the code and its dual yield IrOAs. Moreover, we construct new families of linear IrOAs based on self-dual, Maximum Distance Separable (MDS), and MDS-self-dual codes. Finally, we establish bounds on the minimum distance and covering radius of IrOAs.

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Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [9]

Bounds for the min- imum distance and covering radius of orthogonal arrays via their distance distributions

Boumova, S., Boyvalenkov, P., Stoyanova, M., (2022). Bounds for the min- imum distance and covering radius of orthogonal arrays via their distance distributions. In Proc. 2022 10th International Workshop on Signal Design and Its Applications in Communications (IWSDA) (pp. 1-5)

work page 2022

[2] [1]

Arikan, E. (2009). Channel polarization: A method for constructing capacity- achieving codes for symmetric binary-input memoryless channels, IEEE Transactions on information Theory, 55, 3051-3073

work page 2009

[3] [2]

Arikan, E. (2010). A survey of Reed-Muller codes from polar coding per- spective, in Proc. 2010 IEEE Information Theory Workshop on Information Theory (ITW 2010, Cairo). IEEE, 1–5

work page 2010

[4] [3]

F., & Key, J

Assmus Jr, E. F., & Key, J. D. (1998). Polynomial codes and finite geometries, Chapter 4 (1269-1343) in Huffman, W. C., Pless, V . (editors) Handbook of coding theory, part 2

work page 1998

[5] [4]

Ball, S. (2020). Maximum Distance Separable Codes. In: A Course in Alge- braic Error-Correcting Codes, 83-103, Compact Textbooks in Mathematics, Birkhäuser, Cham

work page 2020

[6] [5]

J., Hangleiter1, D., & Kang, C., (2024)

Barg, A., Coble1, N. J., Hangleiter1, D., & Kang, C., (2024). Geomet- ric structure and transversal logic of quantum Reed–Muller codes, preprint arxiv.2410.07595

work page arXiv 2024

[7] [6]

Bartoli, D., Giulietti, M., & Platoni, I. (2014). On the covering radius of MDS codes. IEEE Transactions on Information Theory, 61(2), 801-811

work page 2014

[8] [7]

Boumova, S., Ramaj, T., & Stoyanova, M. (2021). On covering radius of or- thogonal arrays, Proc. 17th Intern. Workshop Algebr. Combin. Coding The- ory, Bulgaria, Oct. 11-17, 2020, IEEE Xplore, 23–28

work page 2021