The equal-order correspondence conjecture for polycyclic codes associated with trinomials

Let $t_1(x)=x^n-ax^i-b$ and $t_2(x)=x^n-a'x^{n-i}-b'$ satisfy

\operatorname{Ord}(t_1(x))=\operatorname{Ord}(t_2(x)).

Let $S_1$ be the set of all polycyclic codes of length $n$ over $\mathbb{F}_q$ associated with $t_1(x)$ , and let $S_2$ be the corresponding set for $t_2(x)$ . The equal-order correspondence conjecture. The sets $S_1$ and $S_2$ are in one-to-one correspondence, with corresponding codes equivalent to each other. The source paper states that the original conjecture is incorrect and gives a counterexample, so this claim is refuted in the source context.

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).

1 week ago

Progress summary

Refreshed 5 days ago

Solved

The conjecture is false: a published counterexample shows that equal polynomial order does not guarantee the proposed correspondence.

The conjecture appeared as Conjecture $3.5$ in a $2021$ paper on polycyclic codes associated with trinomials, motivated by computational evidence. It asserted a one-to-one correspondence between the two code families, preserving code equivalence, whenever the defining trinomials have equal order.

Known results

The original equal-order condition was proposed as a conjecture, not proved, in $2021$ .
A corrected theorem replaces equal order by the reciprocal relation $t_2(x)=t_1(0)^{-1}t_1^*(x)$ and establishes the desired correspondence under that stronger hypothesis.

2022 counterexample

The later paper states that Conjecture $3.5$ is incorrect. Over $\mathbb{F}_3$ , it uses $t_1(x)=x^{10}+x^8+1$ and $t_2(x)=x^{10}+x^2+2$ , both of order $156$ , but their associated families contain respectively $18$ and $12$ linear codes; hence the conjectured correspondence cannot exist. The work was listed with $2023$ metadata and later publication information.

Current status (as of August 2026): The original equal-order correspondence conjecture is settled as false, while the stronger reciprocal-polynomial correspondence is the valid replacement.

Sources

No solutions have been posted yet.