Existence of indecomposable self-dual codes at every length

Determine whether, for every even length n and every fixed finite field F_q, there exists a self-dual code of length n over F_q whose associated column points form an arithmetically Gorenstein set, equivalently, whether an indecomposable self-dual code exists for every such length and field.

Background

Section 3 establishes that the column points associated with a self-dual code form an arithmetically Gorenstein set precisely when the code is indecomposable. Section 4 therefore reduces the existence of arithmetically Gorenstein self-associated point sets arising from self-dual codes to the existence of indecomposable self-dual codes of prescribed length over a prescribed finite field.

The paper proves only an asymptotic result: for sufficiently large lengths, almost all self-dual codes over any fixed finite field are indecomposable. It also notes that no indecomposable binary self-dual codes exist for lengths 4, 6, and 10, but does not determine the complete set of lengths for which indecomposable self-dual codes exist over each finite field.

References

The next question is whether there exist self-dual codes whose columns points define an arithmetically Gorenstein set for any given length $n$ over a fix finite field $F_q$, or equivalently by Corollary \ref{corollary: mainequivalence}, whether there are indecomposable self-dual codes for any length $n$.

— A combinatorial description of when a self-associated set of points fails to be arithmetically Gorenstein  (2512.16766 - Rodríguez-Pajares et al., 18 Dec 2025) in Section 4, immediately before Theorem 4.1 (the theorem labelled th: asymptotic)