Construction of DNA codes using $\theta$-skew cyclic codes over $\mathbb{F}_4 + v \mathbb{F}_4$

Jo\"el Kabore; Mohammed Elhassani Charkani

REVIEW 4 major objections 4 minor 14 references

Construction of DNA codes using $\theta$-skew cyclic codes over $\mathbb{F}_4 + v \mathbb{F}_4$

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

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Pith's one-line read For even-length θ-skew cyclic codes over F4+vF4, reversibility as a DNA code is equivalent to the generator polynomial being palindromic; odd-degree generators must be θ-palindromic instead.

desk verdict A modest but genuine extension of the DNA-code-from-skew-cyclic program; the main iff theorems need a stated minimal-degree hypothesis and a real proof of the generator classification before they hold as written. read the letter →

arxiv 2506.01236 v1 pith:QVZJ4BBW submitted 2025-06-02 cs.IT math.IT

Joël Kabore , Mohammed Elhassani Charkani This is my paper · ORCID

classification cs.ITmath.IT MSC 94B0594B6011T71

keywords θ-skewcycliccodesDNAreversiblecomplementpalindromicpolynomialsGraymapF4+vF42-quasi-cyclic

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

This paper gives exact algebraic tests for when θ-skew cyclic codes over the finite ring $\mathbb{F}_4+v\mathbb{F}_4$ are usable as DNA codes, where $\mathbb{F}_4$ is the four-element field and $v^2=v$. Its main result is a generator-level condition: for even code length and even generator degree, the code is a reversible DNA code exactly when its generator polynomial is palindromic; odd-degree generators instead must be θ-palindromic. This turns the DNA reverse and reverse-complement constraints, usually checked codeword by codeword, into a single symmetry condition on one polynomial. The paper further shows that Gray images of these codes are 2-quasi-cyclic codes over $\mathbb{F}_4$, giving DNA codes of doubled length with the same reversibility.

What carries the argument

The load-bearing object is the correspondence $\Phi$ from elements of $R$ to DNA 2-bases, built from the Gray map $\varphi(a+bv)=(a+b,a)$, which sends the 16 elements of $R$ to the 16 two-letter DNA words over $\{A,C,G,T\}$. The identity that carries the argument is $\Phi(c)^r=\Phi(\theta(c)^r)$: reversing the DNA word attached to a codeword is the same as applying the order-2 automorphism $\theta(a+bv)=a+b(1+v)$ to each coefficient and reversing their order. On the generator side, a palindromic polynomial is one whose coefficient sequence reads the same forward and backward, and a θ-palindromic polynomial satisfies $a_i=\theta(a_{t-i})$; these are exactly the symmetry conditions that make the reverse of every generated codeword lie in the code. The skew shift $\sigma_\theta(a_0,a_1,\dots,a_{n-1})=(\theta(a_{n-1}),\theta(a_0),\dots,\theta(a_{n-2}))$ is what connects these polynomial symmetries to the DNA reverse operation.

What would settle it

Compute, for a small even $n$, all monic right divisors $g(x)$ of $x^n-1$ in $R[x,\theta]$ with even degree, form $C=\langle g(x)\rangle$, and test whether $\Phi(C)$ is closed under reverse. Theorem 2 predicts closure exactly for palindromic $g$; a single non-palindromic $g$ whose code is reversible, or a palindromic $g$ whose code is not, would settle the claim either way. The encoding premise can be tested independently by comparing $\Phi(\theta(c)^r)$ with the actual Watson-Crick reverse of the DNA string $\Phi(c)$ for the 16 letters of Table 1.

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Extended reading notes

Core claim

The central claim is Theorem 2: if $C=\langle g(x)\rangle$ is a θ-skew cyclic code of even length $n$ over $R=\mathbb{F}_4+v\mathbb{F}_4$ and $\deg(g(x))$ is even, then $C$ is a reversible DNA code if and only if $g(x)$ is a palindromic polynomial. When the degree is odd, reversibility holds exactly when $C$ is generated by a θ-palindromic polynomial (Theorem 3). For odd code length, palindromic or θ-palindromic generators are sufficient, and reversibility forces the code to be cyclic with a palindromic generator in $\mathbb{F}_4[x]$ (Theorem 4). Codes generated by $vg_1(x)$ or $(v+1)g_1(x)$ cannot be reversible under the stated parity conditions (Theorem 5), and reversible-complement DNA codes are precisely the reversible ones whose code also contains the all-one codeword $1+x+\cdots+x^{n-1}$ (Corollary 1).

Load-bearing premise

The load-bearing premise is that reversing the two-letter DNA word attached to a codeword is faithfully modeled by the algebraic rule $\Phi(c)^r=\Phi(\theta(c)^r)$; if this identification does not match real DNA strand behavior, the palindromic test governs algebraic strings rather than usable DNA codes.

Editorial extensions

If this is right

Any even-length θ-skew cyclic code with an even-degree palindromic generator is automatically reversible as a DNA code, so no per-codeword checking is needed.
Odd-degree generators require the stronger θ-palindromic symmetry, so the same reversibility can be obtained at additional lengths whenever such divisors exist.
Reversible-complement DNA codes are exactly the reversible ones whose code also contains the all-one word, which is a checkable condition on the generator.
Generators of the form $vg_1(x)$ or $(v+1)g_1(x)$ cannot produce complement DNA codes at all, and cannot be reversible under the parity conditions of Theorem 5.
Gray images give 2-quasi-cyclic DNA codes over $\mathbb{F}_4$ of length $2n$ with Lee-to-Hamming distance preserved, yielding DNA codes of doubled length.

Reading between the lines

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

The palindromic test is purely algebraic and computable: one could enumerate right divisors of $x^n-1$ in $R[x,\theta]$ for small $n$ and filter for palindromic or θ-palindromic coefficient sequences, obtaining candidate DNA-code libraries without exhaustively testing codewords.
The paper does not address Hamming-distance, GC-content, or melting-temperature constraints; the palindromic criterion could be combined with those constraints to build more realistic DNA-code libraries, but that step is not taken here.
The same θ-automorphism trick may transfer to other rings of the form $\mathbb{F}_q+v\mathbb{F}_q$ with $v^2=v$, provided a Gray map and a DNA 2-base table satisfying the reverse identity exist; the paper does not claim this extension.

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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

Colleague,

The paper is a narrow but real extension of the DNA-code construction program: it works out palindromic and θ-palindromic generator conditions for reversible and reversible-complement θ-skew cyclic codes over R=F4+vF4, using the automorphism θ(a+bv)=a+b(1+v) rather than the ψ used by Bayram et al. The parity split (even length/even degree vs odd degree, and the odd-length reduction to ordinary cyclic codes) is handled carefully, and the 'if' directions—palindromic or θ-palindromic generators do give reversible DNA codes—are sound. The Gray image result (Φ(C) is permutation-equivalent to a 2-quasi-cyclic code) is a nice, correct observation. The DNA encoding via Table 1 and the identity Φ(c)^r = Φ(θ(c)^r) checks out.

The soft spots are real and need to be fixed before the main theorems can be trusted as stated. First, the 'only if' proofs of Theorems 2–4 invoke 'minimality of deg(g)' without ever stating that g is a minimal-degree generator. As written, C=⟨g⟩ does not imply g has minimal degree, so the difference-of-degrees argument can just produce a nonzero codeword of smaller degree; the iff statements are not established. Adding 'where g is a minimal-degree generator' (or 'the generator from Theorem 1') fixes this, but it has to be said. Second, the classification in Theorem 1 is load-bearing and is not proved; deferring to [1], which is over F2+vF2, is not enough, because the unit group and zero-divisor structure of F4+vF4 differ and the automorphism is different. The authors should either prove it or carefully state the transfer. Third, the complement definition in Definition 2.1 is garbled (it defines uc = u); the later usage suggests the intended operation is x ↦ x+1, which works, but the typo matters. Minor issues: the claim that θ is 'more suitable' than ψ is unsupported, and the examples are asserted rather than verified (no code, no check that the listed generators really divide x^n−1).

Bottom line: this is a paper for people working on DNA codes over non-chain rings. The central idea is likely correct after a modest revision, but as it stands the main theorems are conditional. I'd send it to peer review—a good referee can pin down the minimality hypothesis and the Theorem 1 import—but I wouldn't cite Theorem 2 in its current form.

Referee Report

4 major / 4 minor

Summary. The paper studies θ-skew cyclic codes over the ring R = F4 + vF4 with v^2 = v and the order-2 automorphism θ(a + bv) = a + b(1 + v). It states a structure theorem for θ-skew cyclic codes over R, claims exact palindromic and θ-palindromic characterizations of reversible DNA codes (Theorems 2–4), gives non-reversibility conditions for codes generated by zero-divisor polynomials (Theorem 5), derives reversible-complement criteria (Corollaries 1–2), and shows that the Gray image of such a code is a 2-quasi-cyclic code (Theorem 6). Worked examples for lengths 6, 10, and 12 are included.

Significance. If the algebraic classification and the DNA encoding were correct, the palindromic generator test would be an elegant and practically useful exact criterion for constructing reversible DNA codes. The paper appears to be the first to apply this particular automorphism θ to F4 + vF4 in the DNA-coding context, and it provides explicit examples plus a clean Gray-image connection. However, the central DNA results rely on an incorrect identity that does not represent the usual DNA reverse operation, and the proofs also depend on unstated minimal-degree hypotheses and on an unproved structure theorem transferred from a different ring. As it stands, the main contribution is not supported.

major comments (4)

[Section 4 (paragraph before Definition 4.2)] The identity Φ(c)^r = Φ(θ(c)^r) is false for the Gray map φ(a + bv) = (a + b, a). Since θ(a + bv) = a + b(1 + v), one has φ(θ(r)) = (a, a + b), which is the coordinate swap of φ(r) = (a + b, a). For c = (v, 0), Φ(c) = TAAA, the usual reverse is AATA, while θ(c)^r = (0, 1 + v) gives Φ(θ(c)^r) = AAAT. The proofs of the "if" directions of Theorems 2, 3, and 4 all use this identity to identify the reverse of a codeword with an element of C, so those directions do not establish DNA reversibility. The operation actually characterized is c ↦ θ(c^r), not the reverse constraint of Definition 4.2.
[Theorems 2, 3, and 4 (proofs)] Each of these theorems states only that C = ⟨g(x)⟩. The converses invoke "by minimality of deg(g)" (Theorem 2) and "by minimality of the degree of ~g" (Theorem 4) after constructing an element of C of degree less than t. Without an explicit hypothesis that g is a minimal-degree generator, or a proof that the given g is the canonical minimal generator supplied by Theorem 1, that element need not vanish. The coefficient equalities ai = a_{t−i} (or their θ-analogues) therefore do not follow from the stated assumptions. This missing hypothesis is load-bearing for the iff claims.
[Section 3, Lemma 2 and Theorem 1] Lemma 2 and Theorem 1 are not proved; their proofs are deferred to [1], which treats F2 + vF2. The automorphism θ here fixes F4 pointwise and swaps v and 1 + v, but the unit group and zero-divisor structure of F4 + vF4 differ from those of F2 + vF2. Since Theorem 2(ii), Theorem 5, and the forms g = vg1 or g = (v + 1)g1 all depend on this classification, a self-contained proof or a precise transfer argument is required. The current deferred proof does not establish the structure theorem for the ring and automorphism used in this paper.
[Example 5.1 and Theorem 2(ii)] Example 5.1 claims that C = ⟨v(x^4 + x^2 + 1)⟩, of length 6, is a reversible DNA code. Taking the generator as a codeword gives coefficient sequence (v, 0, v, 0, v, 0), so Φ(c) = TA AA TA AA TA AA. Its usual reverse is AA TA AA TA AA TA, which is not in C: every codeword of C is v times an F4[x]-codeword, so every DNA word in Φ(C) has second nucleotide A in each 2-base block, whereas the reverse has first nucleotide A in each block. This concrete contradiction confirms that the operation characterized in the paper is not the DNA reverse used in Definition 4.2.

minor comments (4)

[Section 2, Lemma 1] The formula λ^{-1} = a^{-1} + b^2 v for units of R is stated without proof. It is used in Theorem 3 to determine the possible values of a0, so a short verification should be included.
[Proof of Theorem 2] The displayed expression x^{t+1}g^r(x) = 1 + a_{t−1}x + ⋯ + a_0x^t omits the θ on the coefficients; with the definition of g^r it should be θ(a_{t−1}), …, θ(a_0). The subsequent coefficient comparison needs to be rewritten accordingly.
[Section 2, complement definition] The definition "the complement of u by u^c = (u0, u1, …, u_{n−1})" appears identical to u; the text should clarify that the complement operation is applied coordinatewise (e.g., adding 1 in F4 to each nucleotide).
[Examples 4.1 and 4.2] The examples assert that the listed polynomials are right divisors of x^n − 1 over R[x, θ] without showing a verification. A short computation or a reference to a checked divisor would make the illustrations easier to trust.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the generator-level DNA-code characterizations are derived from the stated skew-shift structure and an external generator classification, not from their own conclusions.

full rationale

The paper's central claims are conditional algebraic equivalences: given a θ-skew cyclic code C=<g(x)> satisfying divisibility and degree hypotheses, C is a reversible DNA code iff g(x) is palindromic (or θ-palindromic). These equivalences are proved from the skew-shift action, the formal identity Φ(c)^r=Φ(θ(c)^r), and the generating-ideal structure. None of these inputs is the target condition: the palindromic property of the generator is not assumed in the definition of reversibility, and the generator classification in Theorem 1 is deferred to the external reference [1], not to the present authors' prior work. The identity Φ(c)^r=Φ(θ(c)^r) is a designed property of the Table 1 correspondence, and even if that modeling choice is biologically debatable, it is not a circular reduction of the theorems to their conclusions. The only potentially load-bearing unproven point is the invocation of 'minimality of deg(g)' in the converses of Theorems 2-4, since those theorem statements do not explicitly require g to be a minimal-degree generator; that is a correctness/rigor gap rather than a circular step, because the minimality claim is not the palindromic conclusion and is not obtained by assuming it. The self-citation [13] appears only in a background survey of skew cyclic codes over non-chain rings and supports no central premise. No fitted parameter, no data subset, and no 'prediction' derived from its own input appear in the paper. The construction is therefore self-contained as an algebraic derivation, notwithstanding the external theorem-dependency and the minimal-degree gap.

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

No fitted numbers or invented physical entities. The core burden is the deferred generator classification from [1] and the assumed DNA encoding through the Gray map and θ; the rest is standard algebra over F4+vF4.

assumptions (4)

domain assumption Theorem 1 generator classification for θ-skew cyclic codes over R, i.e., every such code is generated by a divisor of x^n-1 in a specific way
Stated with proof 'similar to Corollary 5 in [1]'; all DNA-code theorems assume this structure.
domain assumption The Gray-map correspondence in Table 1 represents DNA reverse and reverse-complement operations through θ, in particular Φ(c)^r=Φ(θ(c)^r)
Section 4, after Definition 4.1; the biological validity of the constructed DNA codes depends on this encoding.
standard math The ring R is identified with F4×F4 via v ↦ (1,0), and units, inverses and the automorphism θ are treated under this decomposition
Used throughout, e.g., Lemma 1 and Theorem 3; standard CRT.
domain assumption The complement of a codeword is coefficient-wise addition of 1 in R
Used in Corollary 1 via c^c(x)+c(x)=1+x+...+x^{n-1}; the definition in the text is mistyped.

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Pith. "Pith review of Construction of DNA codes using $\theta$-skew cyclic codes over $\mathbb{F}_4 + v \mathbb{F}_4$." pith.science (2026). https://pith.science/paper/QVZJ4BBW

@misc{pith2026250601236,
  author       = {Pith},
  title        = {Pith review of: Construction of DNA codes using $\theta$-skew cyclic codes over $\mathbbF_4 + v \mathbbF_4$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVZJ4BBW}},
  note         = {Machine review of arXiv:2506.01236}
}

abstract

In this paper, we investigate $\theta$-skew cyclic codes over the ring $R= \mathbb{F}_4 + v \mathbb{F}_4$, where $v^2=v$ and $\theta$ is a non-trivial automorphism over $\mathbb{F}_4 + v \mathbb{F}_4$. This allows us to describe DNA code over this ring by characterizing $\theta$-skew cyclic reversible DNA codes and $\theta$-skew cyclic reversible complement DNA codes. We also explore the Gray images of $\theta$-skew cyclic codes.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

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Bhardwaj and M

S. Bhardwaj and M. Raka, Skew constacyclic codes over a non-chain ring over Fq[u, v]/ < f(u), g(v), uv− vu >, Appl. Algebra Eng. Com- mun. Comput. 31 (2020) 173-194

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Pith tools

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

[1] [1]

Abualrub, N

T. Abualrub, N. Aydin and P. Seneviratne, On θ-cyclic over F2 + vF2, Aust. J. Com. 54 (2012) 115-126

work page 2012

[2] [2]

Abualrub, A

T. Abualrub, A. Ghrayeb and X. N. Zeng Construction of cyclic codes over GF (4) for DNA computing , J. Franklin Inst. 343 (2006) 448-457

work page 2006

[3] [3]

L. M. Adleman, Molecular computation of solutions to combinatorial problems, Science, 266 (1994) 1021-1024

work page 1994

[4] [4]

Boneh, C

D. Boneh, C. Dunworth and R. J. Lipton, Breaking DES using a molec- ular computer, in DNA Based Computer, eds. R. J. Lipton and E. B. Baum, Ser. Discr. Math. Theor. Comp. Sci., vol. 27 (DIMACS Work- shop, Princeton 1996), pp. 37-66

work page 1996

[5] [5]

Bayram, E

A. Bayram, E. S. Oztas and I. Siap, Codes over F4 + vF4 and some DNA applications, Des. Codes Cryptogr. 80 (2016) 379-393

work page 2016

[6] [6]

Bhardwaj and M

S. Bhardwaj and M. Raka, Skew constacyclic codes over a non-chain ring over Fq[u, v]/ < f(u), g(v), uv− vu >, Appl. Algebra Eng. Com- mun. Comput. 31 (2020) 173-194

work page 2020

[7] [7]

Boucher, W

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes , Appl. Algebra Eng. Commun. Comput. 18 (2007) 379-389

work page 2007

[8] [8]

A. G. D’yachkov, P. A. Vilenkin, I. K. Ismagilov, R. S. Sarbaev, A. Macula, D. Torney, and S. White, On DNA Codes , Problems Inform. Transmission 41 (2005) 349-367

work page 2005