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arxiv: 2605.12097 · v1 · submitted 2026-05-12 · 💻 cs.IT · math.IT

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On the Hamming Distance and LCD Properties of Binary Polycyclic Codes and Their Duals

Sujata Bansal , Pramod Kumar Kewat
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Pith reviewed 2026-05-13 04:19 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords binary polycyclic codesHamming distanceLCD codesself-reciprocal trinomialsreversible codesdual distanceoptimal linear codes
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The pith

Binary polycyclic codes from powers of the trinomials x to the 2 times 3 to the v plus x to the 3 to the v plus 1 have exact Hamming distances, are reversible, and are LCD codes in some cases.

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

The paper studies binary polycyclic codes formed by powers of irreducible polynomials over GF(2). It determines the full algebraic structure of these codes and derives general bounds plus exact values for their minimum Hamming distances and for the distances of their Euclidean duals. Special focus falls on the family of codes tied to powers of the self-reciprocal irreducible trinomial x to the 2 times 3 to the v plus x to the 3 to the v plus 1 for nonnegative integers v. For this family the exact distance is computed for every code, reversibility is proved, and the codes are shown to be linear complementary dual codes under explicit conditions on the exponent. The constructions produce optimal and LCD-optimal binary linear codes of various lengths, and a conjecture is offered that all higher-power members of the family are LCD codes.

Core claim

The central claim is that every binary polycyclic code associated with a positive power of the self-reciprocal irreducible trinomial x^{2·3^v} + x^{3^v} + 1 (v ≥ 0) has a completely determined minimum Hamming distance, is reversible, and is an LCD code whenever the exponent satisfies certain arithmetic conditions; moreover, the dual codes inherit explicit distance results, and the whole family yields optimal binary linear codes including at lengths larger than those previously known.

What carries the argument

The polycyclic code generated by a power of the self-reciprocal irreducible trinomial x^{2·3^v} + x^{3^v} + 1, whose roots and factorization over GF(2) fix the minimum distance and the dual structure.

Load-bearing premise

The trinomials x^{2·3^v} + x^{3^v} + 1 remain irreducible and self-reciprocal over the binary field for every nonnegative integer v, so that their powers produce well-defined cyclic-like factorizations without extra roots or repeated factors.

What would settle it

Computation of the weight distribution for the code of length 3^{v+1} generated by (x^{2·3^v} + x^{3^v} + 1)^k for small v and k, showing a nonzero codeword whose weight is strictly smaller than the claimed exact distance.

read the original abstract

Polycyclic codes offer a natural generalization of cyclic codes and provide a broader algebraic framework for constructing linear codes with good parameters. In this paper, we study binary polycyclic codes associated with powers of irreducible polynomials. We first determine their complete algebraic structure and then develop general results on their minimum Hamming distance, including several exact values and bounds. We also examine the Euclidean duals of these codes and derive corresponding results on the Hamming distance of the dual codes. Furthermore, we study the LCD (linear complementary dual) properties of binary polycyclic codes, establish necessary and sufficient conditions for such codes to be LCD codes, and construct several families of binary LCD codes. Our constructions also yield many optimal and LCD optimal binary linear codes, including codes of larger lengths. We then focus on binary polycyclic codes associated with powers of the self-reciprocal irreducible trinomials $x^{2\cdot3^v}+x^{3^v}+1$, where $v\geq0$. For this class, we determine the exact Hamming distance of all such codes and show that these codes are reversible. Moreover, we show that these codes are LCD codes in certain cases. In addition, we propose a conjecture asserting that all binary polycyclic codes associated with $\big(x^{2\cdot3^v}+x^{3^v}+1\big)^{2^\mathcal{T}}$, where $v\geq 0$ and $\mathcal{T}\geq1$, are LCD codes. These results demonstrate that binary polycyclic codes form a rich source of structured codes with strong distance, duality, reversibility, and LCD properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines binary polycyclic codes generated by powers of irreducible polynomials over GF(2). It first derives their algebraic structure via the factorization of the associated polynomial x^n - f(x), then obtains general bounds and several exact values for the minimum Hamming distance of the codes and their Euclidean duals. Necessary and sufficient conditions for the codes to be LCD are established, and explicit constructions of optimal and LCD-optimal binary linear codes are given, including families of larger lengths. The central focus is the infinite family of polycyclic codes associated with positive powers of the self-reciprocal irreducible trinomial x^{2·3^v} + x^{3^v} + 1 (v ≥ 0); for this family the paper asserts exact Hamming distances, proves reversibility, shows the LCD property holds in certain cases, and proposes a conjecture that all such codes with exponent 2^T (T ≥ 1) are LCD.

Significance. If the exact-distance claims are confirmed, the work supplies new infinite families of binary linear codes whose parameters are completely determined, together with reversible and LCD constructions that generalize known cyclic-code results. The LCD-optimal codes and the conjecture on higher-power LCD behavior would be useful for applications in cryptography and error correction that exploit complementarity or reversibility. The general algebraic-structure and duality results also broaden the toolkit for polycyclic codes beyond the cyclic case.

major comments (2)
  1. [§5] §5 (the family associated with powers of x^{2·3^v} + x^{3^v} + 1): the assertion that the minimum Hamming distance is exactly determined (rather than merely bounded) for every positive integer exponent k on the trinomial requires an explicit minimum-weight codeword or weight-enumerator argument once root multiplicity exceeds one. The root-counting/BCH-style lower bound is correctly obtained from the factorization in the extension field, but the manuscript does not exhibit a codeword attaining the bound for arbitrary v and k > 1; this step is load-bearing for the central “exact distance” claim.
  2. [§6] §6 (LCD properties): the necessary-and-sufficient condition for a polycyclic code to be LCD is stated in terms of the generator polynomial and the reciprocal; however, when the generator is a proper power of the trinomial the multiplicity affects the intersection with the dual, and the proof sketch does not explicitly verify that the intersection remains trivial for the claimed parameter ranges.
minor comments (2)
  1. [Preliminaries] The notation for the exponent 2^T (with script T) is introduced only in the abstract and conjecture; define it explicitly in the preliminaries or at the start of §5.
  2. [§5] Several distance tables list parameters for small v but omit the corresponding generator polynomials; adding one column or a short appendix would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, indicating the changes we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§5] §5 (the family associated with powers of x^{2·3^v} + x^{3^v} + 1): the assertion that the minimum Hamming distance is exactly determined (rather than merely bounded) for every positive integer exponent k on the trinomial requires an explicit minimum-weight codeword or weight-enumerator argument once root multiplicity exceeds one. The root-counting/BCH-style lower bound is correctly obtained from the factorization in the extension field, but the manuscript does not exhibit a codeword attaining the bound for arbitrary v and k > 1; this step is load-bearing for the central “exact distance” claim.

    Authors: We agree that an explicit minimum-weight codeword is needed to rigorously establish exactness for k > 1. In the revised version we will insert a direct construction: for the code generated by g(x)^k with g(x) = x^{2·3^v} + x^{3^v} + 1, the vector corresponding to the polynomial g(x)^{k-1} · (x + 1) lies in the code and has Hamming weight exactly equal to the BCH-type lower bound obtained from the root multiplicities in the extension field. This construction works uniformly for all v ≥ 0 and k > 1, thereby confirming that the distance is attained and the claim is exact rather than merely a bound. revision: yes

  2. Referee: [§6] §6 (LCD properties): the necessary-and-sufficient condition for a polycyclic code to be LCD is stated in terms of the generator polynomial and the reciprocal; however, when the generator is a proper power of the trinomial the multiplicity affects the intersection with the dual, and the proof sketch does not explicitly verify that the intersection remains trivial for the claimed parameter ranges.

    Authors: We thank the referee for highlighting the need for explicit verification when the generator has multiplicity greater than one. In the revision we will expand the proof of the LCD criterion to treat the case g(x)^k explicitly. We will show that gcd(g(x)^k, g(x^{-1})^k) = 1 still holds for the parameter ranges stated in the paper (by using the self-reciprocal property of g(x) and the fact that g(x) remains coprime to its reciprocal even after raising to the k-th power). This will confirm that the intersection of the code with its dual is trivial, completing the argument. revision: yes

Circularity Check

0 steps flagged

No circularity: algebraic derivations are self-contained

full rationale

The paper's core results on algebraic structure, exact Hamming distances, reversibility, and LCD properties for polycyclic codes generated by powers of the given self-reciprocal trinomial follow from explicit factorization of the generator polynomial, root multiplicities in extension fields, and standard BCH-style bounds combined with explicit codeword constructions or weight enumerators. These steps rely on field arithmetic and polynomial division that are independent of the target distance or LCD conclusions; no parameter is fitted to data and then relabeled as a prediction, no uniqueness theorem is imported via self-citation, and no ansatz is smuggled. The final conjecture is labeled as such rather than asserted as proven. The derivation chain therefore remains non-circular and externally verifiable via direct computation for small v.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of finite fields and polynomial rings over GF(2); no free parameters are introduced or fitted, no new entities postulated, and axioms are background algebra rather than ad-hoc to this work.

axioms (2)
  • standard math Finite fields GF(2^m) are fields with characteristic 2 and every element satisfies x^{2^m} = x.
    Invoked implicitly when discussing roots of irreducible polynomials and code definitions over binary alphabets.
  • standard math Irreducible polynomials generate maximal ideals in the polynomial ring, allowing quotient rings to be fields.
    Used to define the algebraic structure of polycyclic codes associated with powers of irreducibles.

pith-pipeline@v0.9.0 · 5591 in / 1557 out tokens · 82547 ms · 2026-05-13T04:19:48.163195+00:00 · methodology

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Reference graph

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