Optimal p-ary cyclic codes with two zeros

Xiwang Cao; Yan Liu

REVIEW 3 major objections 2 minor 25 references

Optimal p-ary cyclic codes with two zeros

T0 review · 3 major / 2 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper constructs four families of optimal p-ary cyclic codes with two zeros, all attaining the sphere-packing bound with parameters $[n,n-2m,4]$.

desk verdict Three of the four families look sound, but the advertised v=p^k+1 class is unproved: its congruence check fails for every p>3. read the letter →

arxiv 1908.03070 v1 pith:ZDEWC3R4 submitted 2019-08-08 cs.IT math.IT

Yan Liu , Xiwang Cao This is my paper · ORCID

classification cs.ITmath.IT MSC 94B1511T71

keywords cycliccodesoptimalp-arysphere-packingboundtwozeroscyclotomiccosetsminimumdistance4finitefields

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

The paper sets out to construct new optimal cyclic error-correcting codes over prime fields, using the 'two zeros' format where the generator polynomial is a product of two minimal polynomials. It claims that for each admissible choice of a prime $p$ and integer $m>2$, with $n=2(p^m-1)/(p-1)$, there are cyclic codes over $\mathbb{F}_p$ with parameters $[n,n-2m,4]$ that attain the sphere-packing bound and are therefore optimal. The proof runs through two theorems: one for exponents of the form $v=p^k+1$, and one for exponents $v$ solving the congruence $(p^t-1)v\equiv p^s-p^h\pmod{p^m-1}$, from which three explicit families follow. A reader should care because optimal cyclic codes combine the strongest possible error correction for their size with the efficient algebraic decoding that cyclic structure provides.

What carries the argument

The central object is the cyclic code $C_p(1,v)$, whose generator polynomial is the product $m_1(x)m_v(x)$ of two minimal polynomials; this is what 'two zeros' means. The device that carries the proof is Lemma 2.4, a sufficient criterion taken from [15]: $C_p(1,v)$ is an optimal $[n,n-2m,4]$ code if $v\notin C_1$, the cyclotomic coset of $v$ has length $m$, $\gcd(v-1,n)=1$, $v\equiv1\pmod{(p-1)/2}$, and the equations $(x+\alpha)^v\pm(x^v+\alpha)=0$ have no solution $x\in\Pi\setminus\{-1\}$ for any $\alpha\in\mathbb{F}_p^*$. The new work is to verify that no-solution condition for specific $v$: Theorem 3.1 does it for $v=p^k+1$ by raising a hypothetical solution to the $p^k-1$ power, and Theorem 3.4 does it for solutions of the congruence above by raising to the $p^t-1$ power and reducing the obstruction to $x_0^{p^s-p^h}=1$ or $x_0^{p^h-v}=1$, each excluded by a gcd condition.

What would settle it

For $p=5$, $m=5$, $k=2$, Theorem 3.1 predicts an optimal $[1562,1552,4]$ cyclic code with $v=26$; computing the true minimum distance and finding a nonzero codeword of weight below 4 would disprove the theorem. A more direct check is to search $\Pi\setminus\{-1\}$ and $\alpha\in\mathbb{F}_5^*$ for a solution to $(x+\alpha)^{26}\pm(x^{26}+\alpha)=0$, which the proof asserts cannot exist.

Watch

Extended reading notes

Core claim

The central claim is that the cyclic code $C_p(1,v)$ of length $n=2(p^m-1)/(p-1)$ over $\mathbb{F}_p$ is an optimal $[n,n-2m,4]$ code whenever $v$ satisfies the conditions of Lemma 2.4: $v\notin C_1$, the cyclotomic coset of $v$ has length $m$, $\gcd(v-1,n)=1$, $v\equiv1\pmod{(p-1)/2}$, and the equations $(x+\alpha)^v\pm(x^v+\alpha)=0$ have no solution in $\Pi\setminus\{-1\}$ for any $\alpha\in\mathbb{F}_p^*$. The paper supplies new exponents $v$ that pass this test. Theorem 3.1 shows it for $v=p^k+1$ under $\gcd(m,k)=\gcd(m,p-1)=1$ and $(p-1)/2\mid k$ with $m>2$ odd. Theorem 3.4 shows it for any solution $v$ of $(p^t-1)v\equiv p^s-p^h\pmod{p^m-1}$ with $\gcd(m,t)=\gcd(m,s-h)=1$, $\gcd(m,p-1)\mid2$, plus the additional coprimality and congruence conditions. Corollaries 3.5, 3.7, and 3.9 turn these into explicit families, and the examples give concrete codes, including a $[62,50,4]$ code over $\mathbb{F}_5$, a $[5602,5592,4]$ code over $\mathbb{F}_7$, and a $[728,714,4]$ code over $\mathbb{F}_3$.

Load-bearing premise

The proof leans on Lemma 2.4, taken from [15], which asserts that its listed conditions on the exponent $v$ are enough to force the code to have distance 4 and to meet the sphere-packing bound; the paper does not re-prove that lemma, so a hidden flaw or missing hypothesis there would invalidate the optimality claims for every new family.

Editorial extensions

If this is right

Every admissible pair $(p,m)$ yields a cyclic code with parameters $[2(p^m-1)/(p-1), 2(p^m-1)/(p-1)-2m, 4]$ over $\mathbb{F}_p$ that meets the sphere-packing bound.
Because the codes are cyclic, they keep the efficient algebraic encoding and decoding typical of cyclic codes; minimum distance 4 means every single symbol error is correctable and up to three errors are detectable.
The $p=3$ case of Corollary 3.5 recovers a previously known ternary optimal cyclic code family as a special case and extends it to every odd $m\ge3$ satisfying the stated gcd conditions.
The explicit examples supply generator polynomials for optimal codes over $\mathbb{F}_5$, $\mathbb{F}_7$, and $\mathbb{F}_3$, including a $[62,50,4]$ code over $\mathbb{F}_5$.

Reading between the lines

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

The same exponent-raising method could be tried on other congruences of the form $(p^t-1)v\equiv p^s-p^h$ with shifted exponents or coefficients; the paper does not explore those, and each candidate would still have to pass Lemma 2.4's no-solution check.
The paper does not compute the weight enumerators of the new codes or their duals; for earlier two-zero cyclic codes such enumerators have often been tractable, so the new families are natural candidates for that analysis.
A finite-field search over small primes and exponents could test whether the gcd conditions in Theorems 3.1 and 3.4 are also necessary for the no-solution condition to hold, or merely sufficient.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Read the paper quickly: the real content is Theorem 3.4 and its three corollaries. Those appear to be new infinite families of distance-optimal p-ary cyclic codes, and the proofs are reconstructible: the Frobenius manipulation reduces the no-solution condition to gcd statements, and the gcd arithmetic checks out for the cases used. The p=3 special case of Corollary 3.5 generalizes the Ding–Helleseth result, which is a legitimate extension. That part deserves referee time.

The first class, Theorem 3.1, does not hold as stated. The proof verifies condition 2 of Lemma 2.4 with 'v≡k+1≡1 (mod (p−1)/2)'. But v=p^k+1 and p≡1 mod (p−1)/2, so v≡2 mod (p−1)/2, not 1, for any p>3. The stress-test is right: the hypothesis (p−1)/2 | k makes things worse, because v≡2 is never 1 when (p−1)/2>1. In fact Lemma 2.4's exemption of −1 fails: for p=5, m=3, k=2, x=−1, α=3 solves (x+α)^v − (x^v+α)=0 in F_5. So Example 3.2's claimed optimality is unsupported. This is not a minor typo; it is the paper's advertised 'another class'.

Example 3.10 is also broken: with p=3, m=7, n=2186, the claimed parameters [728,714,4] cannot be right, and 'e=86' matches neither the corollary nor n/2+(p^s−1)/(p−1)=1106. The reader's take flagged this as a conditional issue; it is worse than a typo because it suggests the examples were not computed from the displayed formulas.

The citation pattern is honest: Lemma 2.4 comes from [15] and is not re-derived, but that is acceptable for a published criterion. The three corollaries do rest on it, so a referee should verify Lemma 2.4 independently. I did not find circular reasoning.

Bottom line: this is a mixed paper. The three congruence families are probably a real, useful contribution to the p-ary cyclic-code literature; the first family should be withdrawn or corrected, and the examples need recomputation. For that reason I would send it to a serious referee rather than desk reject, but acceptance hinges on the authors fixing Theorem 3.1 and the examples. As submitted, I would not cite it, and I would not teach the first class.

Referee Report

3 major / 2 minor

Summary. This paper proposes constructions of optimal p-ary cyclic codes with two zeros, of parameters [n, n-2m, 4] where n=2(p^m-1)/(p-1). The authors use a sufficient condition from Liao et al. (Lemmas 2.3 and 2.4) and verify it for two families of exponents v: Theorem 3.1 takes v=p^k+1 under the hypotheses gcd(m,k)=gcd(m,p-1)=1 and (p-1)/2 | k; Theorem 3.4 takes any solution v of (p^t-1)v ≡ p^s-p^h (mod p^m-1) satisfying the stated gcd and congruence conditions, from which three corollaries and several examples are derived.

Significance. Optimal two-zero cyclic codes are a well-studied topic, and a genuinely new family with parameters meeting the sphere-packing bound would be a useful contribution. The proof strategy reduces the no-solution condition in Lemma 2.4 to gcd computations, and the gcd arguments in Theorem 3.4 are essentially reconstructible; Examples 3.6 and 3.8 are consistent with the stated formulas. The significance is weakened by a serious error in Theorem 3.1 and by inconsistent numerical examples, so the contribution currently rests on Theorem 3.4 alone.

major comments (3)

[Section 3.1, Theorem 3.1 and its proof] Condition 2 of Lemma 2.4 cannot be satisfied for p>3. In the proof, the congruence 'v≡k+1≡1 (mod (p-1)/2)' is arithmetically wrong: because p≡1 (mod (p-1)/2), v=p^k+1≡2 (mod (p-1)/2), independent of k. Since 2≠1 modulo (p-1)/2 whenever p>3, the theorem proves nothing for p>3 and is at best a p=3 statement. Example 3.2, which claims p=5, is therefore outside the stated hypothesis.
[Example 3.2] The numerical data in Example 3.2 are inconsistent with the formulas: for p=5, m=3 one has n=62 and v=5^2+1=26, not 10, and the dimension of C_p(1,v) should be n-2m=56, not 50. The displayed generator polynomial has degree 6, consistent with dimension 56, so the stated parameters [62,50,4] need correction.
[Example 3.10] The data in Example 3.10 are also inconsistent: p=3, m=7 gives n=2186 and, for s=3, v=n/2+(3^3-1)/(3-1)=1106; the printed parameters [728,714,4] correspond to n=728, and the symbol 'e=86' is undefined. This example should be recalculated or replaced.

minor comments (2)

[Remark after Lemma 2.3] The remark asserts that conditions 1) and 2) imply v is even and p≡3 mod 4; this implication is used in the proof of Lemma 2.4 but not proved there. It would be helpful to spell out the argument.
[Throughout] There are several typographical errors in the text and references ('monomilas', 'minimun distance', 'Intertional'); these should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new code families are verified against an external optimality criterion, not defined by it.

full rationale

The paper's central derivations (Theorem 3.1, Theorem 3.4, and the three corollaries) are ordinary sufficient-condition verifications. Lemma 2.4, the load-bearing optimality criterion, is taken from the external reference [15]; it is not derived from the paper's own target results, and the paper's proof of Lemma 2.4 is only a reduction to Lemma 2.3, also from [15], with an elementary check that -1 cannot solve the displayed equations under conditions 1 and 2. Neither theorem fits a parameter and then renames it a prediction, nor does any step define the constructed exponent v in terms of the code's minimum distance. The self-citation [21] appears only in the introduction as related work and is not load-bearing. A possible arithmetical flaw in the verification of condition 2 in Theorem 3.1 would be a correctness defect, not a circular dependency, so it does not raise the circularity score. Overall, the derivation is self-contained relative to its external hypotheses.

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

No fitted constants enter; the construction parameters m,k,t,s,h are free integers in the stated families. The proofs borrow three lemmas from [15] and the sphere-packing criterion; these are the external axioms the central claim rests on.

assumptions (4)

domain assumption Lemma 2.1 ([15]): if gcd(j,n)=d with 1≤d≤2(p+1), then the cyclotomic coset C_j has length m.
Used to conclude C_v has length m when gcd(v,n)=1 or 2 in Theorems 3.1 and 3.4.
domain assumption Lemma 2.2 ([15]): the minimum distance of Cp(u,v) is at least 3 if gcd(v-u,n)=1.
Part of the inherited optimality framework for cyclic codes with two zeros.
domain assumption Lemma 2.3 ([15]): the three conditions (gcd(v-1,n)=1, v≡1 mod (p-1)/2, and no-solution condition) imply Cp(1,v) is optimal with parameters [n,n-2m,4].
Central optimality criterion; Lemma 2.4 is a rewording with the same content plus a proof that -1 is not a solution.
standard math Sphere-packing bound: a code with parameters [n,n-2m,4] is optimal because distance 5 would violate the bound.
Used to justify the word optimal in every theorem and corollary.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Optimal p-ary cyclic codes with two zeros." pith.science (2026). https://pith.science/paper/ZDEWC3R4

@misc{pith2026190803070,
  author       = {Pith},
  title        = {Pith review of: Optimal p-ary cyclic codes with two zeros},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZDEWC3R4}},
  note         = {Machine review of arXiv:1908.03070}
}

read the original abstract

As a subclass of linear codes, cyclic codes have efficient encoding and decoding algorithms, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, we give a general construction of optimal p-ary cyclic codes which leads to three explicit constructions. In addition, another class of p-ary optimal cyclic codes are presented.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

25 extracted references · 24 canonical work pages

[15]

D. Liao, X. Kai, S. Zhu, P. Li , A class of optimal cyclic codes with t wo zeros, IEEE Commun.Lett. , DOI: 10.1109/LCOMM.2019.2921330

arXiv 2019
[1]

Charpin, A

P. Charpin, A. Tiet¨ av¨ ainen, V. Zinoviev, On binary cyclic codes w ith distance three, Probl. Inf. Transm. , 33 (1997), 3-14. 6

work page 1997
[2]

Charpin, A

P. Charpin, A. Tiet¨ av¨ ainen, V. Zinoviev, On the minimum distances of non-binary cyclic codes, Des. Codes Crypogr. , 17 (1999), 81-85

work page 1999
[3]

Carlet, C

C. Carlet, C. Ding, J. Yuan, Linear codes from highly nonlinear fun ctions and their secret sharing schemes, IEEE Trans. Inf. Theory , 51, no. 6 (2005), 2089-2102

work page 2005
[4]

C. Ding, T. Helleseth, Optimal ternary cyclic codes from monomials , IEEE Trans. Inf. Theory, 59, no. 9 (2013), 5898-5904

work page 2013
[5]

C. Ding, S. Ling, A q-polynomial approach to cyclic codes, Finite Fields Appl. , 20 (2013), 1-14

work page 2013
[6]

C. Ding, C. Li, N. Li, Z. Zhou, Three-weight cyclic codes and their w eight distri- butions, Discrete Math., 339, no. 2 (2016), 415-427

work page 2016
[7]

C. Ding, Y. Liu, C. Ma, L. Zeng, The weight distributions of the dua ls of cyclic codes with two zeros, IEEE Trans. Inf. Theory , 57, no. 12 (2011), 8000-8006

work page 2011

Show all 25 references

[8]

C. Fan, N. li, Z. Zhou, A class of optimal ternary cyclic codes and t heir duals, Finite Fields Appl. , 37 (2016), 193-202

2016
[9]

W. Fang, J. Wen, F. Fu, A q-polynomial approach to constacyclic codes, Finite Fields Appl. , 47 (2017), 161-182

2017
[10]

Feng, On cyclic codes of length 2 2r − 1 with two zeros whose dual codes have three weights, Des

T. Feng, On cyclic codes of length 2 2r − 1 with two zeros whose dual codes have three weights, Des. Codes Crypogr. , 62 (2012), 253-258

2012
[11]

Huﬀman, V

W. Huﬀman, V. Pless, Foundamentals of error-correcting cod es, Cambridge Uni- versity Press , (2003)

2003
[12]

C. Li, Q. Yue, Weight distributions of two classes of cyclic codes w ith respect to two distinct order elements, IEEE Trans. Inf. Theory , 60, no. 1 (2014), 296-303

2014
[13]

N. Li, C. Li, T. Helleseth, C. Ding, X. Tang, Optimal ternary cyclic codes with minimun distance four and ﬁve, Finite Fields Appl. , 30 (2014), 100-120

2014
[14]

N. Li, Z. Zhou, T. Helleseth, On a conjecture about a class of op timal ternary cyclic codes, Signal Design and its Applications in Communications (IWS DA), 2015 seventh Intertional Workshop on Signal, DOI:10.1109/IWSDA .2015.7458415

2015
[16]

J. Luo, K. Feng, Cyclic codes and sequences form generalized C oulter-Matthews function, IEEE Trans. Inf. Theory , 54, no. 12 (2008), 5345-5353

2008
[17]

C. Ma, L. Zeng, Y. Liu, D. Feng, C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inf. Theory , 57, no. 1 (2011), 397-402

2011
[18]

M. Shi, P. Sol´ e, Optimal p-ary codes from one-weight and two-weight codes over Fp + vF∗ p, J Syst Sci Complex , 28, no. 3 (2015), 679-690

2015
[19]

L. Wang, G. Wu, Several classes of optimal ternary cyclic code s with minimal distance four, Finite Fields Appl. , 40 (2016), 126-137. 7

2016
[20]

Xiong, N

M. Xiong, N. Li, Optimal cyclic codes with generalized Niho-type ze ros and the weight distribution, IEEE Trans. Inf. Theory , 61, no. 9 (2015), 4914-4922

2015
[21]

G. Xu, X. Cao, S. Xu, Optimal p-ary cyclic codes with minimum distance four from monomials, Cryptography and Communications , 8, no. 4 (2016), 541-554

2016
[22]

H. Yan, Z. Zhou, X. Du, A family of optimal ternary cyclic codes f rom the Niho- type exponent, Finite Fields Appl. , 54 (2018), 101-112

2018
[23]

X. Zeng, L. Hu, W. Jiang, The weight distribution of a class of p-ary cyclic codes, Finite Fields Appl. , 16, no. 1 (2010), 126-137

2010
[24]

Y. Zhou, X. Kai, S. Zhu, J. Li, On the minimum distance of negacyc lic codes with two zeros, Finite Fields Appl. , 55 (2019), 134-150

2019
[25]

Z. Zhou, C. Ding, A class of three-weight cyclic codes, Finite Fields Appl. , 25 (2014), 79-93. 8

2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [15]

D. Liao, X. Kai, S. Zhu, P. Li , A class of optimal cyclic codes with t wo zeros, IEEE Commun.Lett. , DOI: 10.1109/LCOMM.2019.2921330

arXiv 2019

[2] [1]

Charpin, A

P. Charpin, A. Tiet¨ av¨ ainen, V. Zinoviev, On binary cyclic codes w ith distance three, Probl. Inf. Transm. , 33 (1997), 3-14. 6

work page 1997

[3] [2]

Charpin, A

P. Charpin, A. Tiet¨ av¨ ainen, V. Zinoviev, On the minimum distances of non-binary cyclic codes, Des. Codes Crypogr. , 17 (1999), 81-85

work page 1999

[4] [3]

Carlet, C

C. Carlet, C. Ding, J. Yuan, Linear codes from highly nonlinear fun ctions and their secret sharing schemes, IEEE Trans. Inf. Theory , 51, no. 6 (2005), 2089-2102

work page 2005

[5] [4]

C. Ding, T. Helleseth, Optimal ternary cyclic codes from monomials , IEEE Trans. Inf. Theory, 59, no. 9 (2013), 5898-5904

work page 2013

[6] [5]

C. Ding, S. Ling, A q-polynomial approach to cyclic codes, Finite Fields Appl. , 20 (2013), 1-14

work page 2013

[7] [6]

C. Ding, C. Li, N. Li, Z. Zhou, Three-weight cyclic codes and their w eight distri- butions, Discrete Math., 339, no. 2 (2016), 415-427

work page 2016

[8] [7]

C. Ding, Y. Liu, C. Ma, L. Zeng, The weight distributions of the dua ls of cyclic codes with two zeros, IEEE Trans. Inf. Theory , 57, no. 12 (2011), 8000-8006

work page 2011