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

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.