TL;DR
This paper investigates the extension property for linear codes over module alphabets, proving a complete extension theorem for symmetrized weight composition and showing limitations for noncyclic socle modules.
Contribution
It introduces a complete extension theorem for symmetrized weight composition and demonstrates the failure of extension properties for certain module alphabets.
Findings
Complete extension theorem for symmetrized weight composition
Extension property fails for module alphabets with noncyclic socle
Limitations of extension properties with arbitrary weight functions
Abstract
A complete extension theorem for linear codes over a module alphabet and the symmetrized weight composition is proved. It is shown that an extension property with respect to arbitrary weight function does not hold for module alphabets with a noncyclic socle.
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