Narrow Results

In this paper, we study codes where the alphabet is a finite Frobenius bimodule over a finite ring. We discuss the extension property for various weight functions. Employing an entirely character-theoretic approach and a duality theory for partitions on Frobenius bimodules, we derive alternative proofs for the facts that the Hamming weight and the homogeneous weight satisfy the extension property. We also use the same techniques to derive the extension property for other weights, such as the Rosenbloom–Tsfasman weight.

When $C\subseteq {𝔽}^n$ is a linear code over a finite field $𝔽$, every linear Hamming isometry of $C$ to itself is the restriction of a linear Hamming isometry of ${𝔽}^n$ to itself, i.e. a monomial transformation. This is no longer the case for additive codes over non-prime fields. Every monomial transformation mapping $C$ to itself is an additive Hamming isometry, but there may exist additive Hamming isometries that are not monomial transformations.

The monomial transformations mapping $C$ to itself form a group $\mathrm{\text{rMon}}(C)$, and the additive Hamming isometries form a larger group $\mathrm{\text{Isom}}(C)$: $\mathrm{\text{rMon}}(C)\subseteq \mathrm{\text{Isom}}(C)$. The main result says that these two subgroups can be as different as possible: for any two subgroups $H_1\subseteq H_2$, subject to some natural necessary conditions, there exists an additive code $C$ such that $\mathrm{\text{rMon}}(C)=H_1$ and $\mathrm{\text{Isom}}(C)=H_2$.

The geometric characterization of the extension property for Cantor-type sets, found in [3], is related to the rate of growth of the values of the discrete logarithmic energies of compact sets that locally form the set.