Some Constacyclic Codes over Finite Chain Rings

For $\lambda$ an $n$-th power of a unit in a finite chain ring we prove that $\lambda$-constacyclic repeated-root codes over some finite chain rings are equivalent to cyclic codes. This allows us to simplify the structure of some constacylic codes. We also study the $\alpha +p \beta$-constacyclic codes of length $p^s$ over the Galois ring $GR(p^e,r)$

MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question.Comment: 12 page

The extension problem for Lee and Euclidean weights

The extension problem is solved for the Lee and Euclidean weights over three families of rings of the form $\Z/N\Z$: $N=2^{\ell + 1}$, $N=3^{\ell + 1}$, or $N=p=2q+1$ with $p$ and $q$ prime. The extension problem is solved for the Euclidean PSK weight over $\Z/N\Z$ for all $N$

MacWilliams' Extension Theorem for rank-metric codes

The MacWilliams' Extension Theorem is a classical result by Florence Jessie MacWilliams. It shows that every linear isometry between linear block-codes endowed with the Hamming distance can be extended to a linear isometry of the ambient space. Such an extension fails to exist in general for rank-metric codes, that is, one can easily find examples of linear isometries between rank-metric codes which cannot be extended to linear isometries of the ambient space. In this paper, we explore to what extent a MacWilliams' Extension Theorem may hold for rank-metric codes. We provide an extensive list of examples of obstructions to the existence of an extension, as well as a positive result.Comment: 12 page