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arxiv: 1907.07107 · v1 · pith:TGST5IHJnew · submitted 2019-07-14 · 💻 cs.IT · math.IT

An efficient method to construct self-dual cyclic codes of length p^s over mathbb{F}_(p^m)+umathbb{F}_(p^m)

Yuan Cao , Yonglin Cao , Hai Q. Dinh , Somphong Jitman This is my paper

Pith reviewed 2026-05-24 21:40 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords self-dual cyclic codesfinite chain ringsKronecker productcombinatorial identitiesexplicit constructioncyclic codes over ringslength p^schain ring codes
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The pith

Self-dual cyclic codes of length p^s over F_{p^m} + u F_{p^m} admit an explicit representation and efficient construction derived from Kronecker product properties over F_p.

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

The paper first derives properties of Kronecker products for a specific class of matrices over the prime field F_p by means of combinatorial identities. These properties are then applied to produce an explicit form for each self-dual cyclic code of length p^s over the chain ring F_{p^m} + u F_{p^m} with u^2 = 0. The same machinery supplies both an exact count of all distinct such codes and a direct procedure to build any chosen one. A reader interested in algebraic coding would see value in replacing exhaustive search or indirect classification with a concrete listing and construction rule that works for any odd prime p and positive integers s and m.

Core claim

Using some combinatorial identities, certain properties are obtained for the Kronecker product of matrices over F_p with a specific type. On that basis an explicit representation and enumeration are given for all distinct self-dual cyclic codes of length p^s over the finite chain ring F_{p^m} + u F_{p^m} (u^2 = 0). An efficient method is also provided to construct every such self-dual cyclic code precisely.

What carries the argument

Kronecker product of matrices over F_p of a specific type, whose properties derived from combinatorial identities map directly to the generators of the self-dual cyclic codes.

If this is right

  • Every self-dual cyclic code of length p^s over the ring possesses a unique explicit representation obtained from the matrix construction.
  • The total number of distinct self-dual cyclic codes of the given length and ring is finite and can be stated by a closed formula.
  • Any chosen self-dual cyclic code can be generated directly by the procedure without enumeration or search.
  • The representation and construction hold uniformly for every odd prime p and every positive integers s and m.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The matrix-product technique may be tested on other lengths or on related rings to see whether similar explicit forms appear.
  • The explicit generators could be used to compute weight distributions or minimum distances for these codes in concrete cases.
  • If the same Kronecker-product identities extend to non-cyclic codes, the method might supply constructions beyond the cyclic case.

Load-bearing premise

The combinatorial identities for the Kronecker product of matrices over F_p with the specific type are sufficient to produce the claimed explicit representation of the codes.

What would settle it

For small values such as p=3, s=1, m=1, compute all self-dual cyclic codes of length 3 over the ring by direct enumeration and check whether their count and generators match the formula and construction given by the method.

read the original abstract

Let $p$ be an odd prime number, $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $s$ a positive integer. Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over $\mathbb{F}_p$ with a specific type. On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length $p^s$ over the finite chain ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$. Moreover, We provide an efficient method to construct every self-dual cyclic code of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ precisely.

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

0 major / 2 minor

Summary. The paper claims that combinatorial identities yield properties of Kronecker products of specific matrices over F_p; these properties are then used to obtain an explicit representation and complete enumeration of all distinct self-dual cyclic codes of length p^s over the chain ring R = F_{p^m} + u F_{p^m} (u^2 = 0), together with an efficient construction method for each such code.

Significance. If the derivation holds, the work supplies a precise, explicit enumeration and a constructive algorithm for an entire family of self-dual cyclic codes over a finite chain ring, which is of interest in algebraic coding theory. The approach via Kronecker-product identities over the prime field is a potentially reusable technique.

minor comments (2)
  1. Abstract, last sentence: 'We' is capitalized after a comma; this is a typographical inconsistency.
  2. The manuscript should include at least one concrete small example (e.g., p=3, s=1, m=1) that lists all enumerated codes and verifies self-duality directly from the generator polynomials or matrices.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful review and the recommendation of minor revision. The referee's summary correctly reflects the main results of the manuscript. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins with combinatorial identities applied to Kronecker products of matrices over F_p, then lifts those properties to an explicit enumeration and construction of self-dual cyclic codes over the chain ring. No equations reduce to their own inputs by definition, no parameters are fitted then relabeled as predictions, and no load-bearing claims rest on self-citations or imported uniqueness theorems. The central steps remain independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract only; no free parameters, axioms, or invented entities are identifiable from the given information.

pith-pipeline@v0.9.0 · 5679 in / 1091 out tokens · 31040 ms · 2026-05-24T21:40:24.216478+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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