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The binary characterization conjecture for isodual polycyclic codes

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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.

References

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
Posted

Progress summary

Refreshed
Claimed 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.