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The binary characterization conjecture for isodual polycyclic codes

Let CC be a polycyclic code over F2\mathbb{F}_2 generated by a polynomial g(x)g(x) satisfying

g(x)∣xn−axi−b.g(x)\mid x^n-ax^i-b.

Binary isoduality characterization conjecture. The code CC is isodual if and only if

g2(x)=xn−axi−b.g^2(x)=x^n-ax^i-b.

The source paper states that this conjecture has a counterexample, showing that the proposed characterization is false.

Sources & referencesView supporting material

Primary source

Minjia Shi, Haodong Lu, Shuang Zhou, Jiarui Xu and Yuhang Zhu, “Equivalence and Duality of Polycyclic Codes Associated with Trinomials over Finite Fields”, arXiv:2204.12433 (2022).

coding-theoryisodual-codespolycyclic-codes

Progress summary

Refreshed
Solved

A 2022 paper gives an explicit example showing that the proposed if-and-only-if test is false, while its sufficient direction remains valid.

The conjecture claims that a binary polycyclic code generated by g(x)g(x) is isodual exactly when g2(x)=xn−axi−bg^2(x)=x^n-ax^i-b. The cited paper settles the conjecture negatively by disproving its necessity direction.

Known results

  • The sufficiency direction is proved: g2(x)=xn−axi−bg^2(x)=x^n-ax^i-b implies isoduality over F2\mathbb{F}_2 (2022).

April 2022 counterexample

Example 4.2 constructs an isodual code over F2\mathbb{F}_2 from g(x)=(x2+x+1)(x4+x+1)2g(x)=(x^2+x+1)(x^4+x+1)^2, with x20+x10+1=(x2+x+1)2(x4+x+1)2(x4+x3+1)2x^{20}+x^{10}+1=(x^2+x+1)^2(x^4+x+1)^2(x^4+x^3+1)^2. It shows that the stated square condition is not necessary, so the proposed characterization is false.

Current status (as of August 2026): The conjecture is settled false; sufficiency is established, necessity is disproved by an explicit counterexample, and no narrower replacement characterization was found in the retrieved sources.

Sources

Solutions 0

No solutions have been posted yet.