The prime-length non-dual-containing conjecture for polycyclic codes

Let $C$ be a polycyclic code with generator polynomial $g(x)$ satisfying

g(x)\mid x^n-ax^i-b,

and suppose that $n$ is prime. Prime-length non-dual-containing conjecture. The code $C$ is not dual-containing. The source paper concludes that there are no nontrivial dual-containing polycyclic codes associated with the relevant trinomials, while the trivial generator $g(x)=1$ is the extreme case.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The prime-length non-dual-containing conjecture for polycyclic codes
Let $C$ be a polycyclic code with generator polynomial $g$ dividing $x^n-ax^i-b$ , where $n$ is prime. Prime-length dual-containing conjecture. The code $C$ is not dual-containing. The source places this assertion after the binary nonexistence conjecture and does not provide a proof or resolution.
source: Nuh Aydin, Peihan Liu and Bryan Yoshino, “Polycyclic Codes Associated with Trinomials: Good Codes and Open Questions”, arXiv:2106.12065 (2021).

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).

Additional references

2 papers in this index state this conjecture (2021–2022). The statement above is taken from the most recent of them; the others are arXiv:2108.03567.

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Progress summary

Refreshed 5 days ago

Solved

The conjecture is false: an explicit example shows that a prime-length code can contain its dual.

Stated as Conjecture $4.11$ in 2021, the conjecture asserts that a polycyclic code with $g(x)\mid x^n-ax^i-b$ cannot be dual-containing when $n$ is prime.

Known results

The 2021 paper posed the assertion as an open conjecture (Conjecture $4.11$ ).
For the separate binary case, Theorem $6.1$ establishes that no nontrivial dual-containing polycyclic codes associated with trinomials exist over $\mathbb{F}_2$ ; this does not imply the general conjecture.

2022 counterexample

A follow-up paper disproved Conjecture $4.11$ : over $\mathbb{F}_3$ , the code generated by $g(x)=x+1$ satisfies $g(x)\mid x^3+x-1$ and is dual-containing, with $C^\perp\subseteq C$ . Since $3$ is prime, this directly settles the conjecture negatively.

Current status (as of August 2026): The conjecture is resolved negatively by the explicit length- $3$ counterexample over $\mathbb{F}_3$ , while the separate binary nonexistence result remains valid.

Sources

No solutions have been posted yet.