Code equivalence characterizes finite Frobenius rings

Code Equivalence Characterizes Finite Frobenius Rings

In this paper we show that finite rings for which the code equivalence theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and López-Permouth.

Polynomial Equivalence of Finite Rings

We prove that Zpn and Zp[t]/(t) are polynomially equivalent if and only if n ≤ 2 or p = 8. For the proof, employing Bernoulli numbers, we provide the polynomials which compute the carry-on part for the addition and multiplication in base p. As a corollary, we characterize finite rings of p elements up to polynomial equivalence.

The Equivalence Problem over Finite Rings

We investigate the computational complexity of deciding whether or not a given polynomial , presented as the sum of monomials, is identically 0 over a ring. It is proved that if the factor by the Jacobson-radical is not commutative, then the problem is coNP-complete.

Witt Theorems for Linear Codes over Finite Frobenius Rings

A Coding-theoretic Characterization of Finite Frobenius Rings

In this paper we show that finite rings for which the extension theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and López-Permouth.