In this paper, we provide a framework for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) from classical additive codes over a finite commutative local Frobenius ring. We give a formula for the minimum number of entanglement qudits required to construct an EAQECC from an additive code over a finite Galois ring. This significantly extends known results for EAQECCs over finite fields.

Let p be a prime, s be a positive integer, and R be a finite commutative chain ring with the characteristic as a power of p. For a unit λ ε R, λ-constacyclic codes of length ps over R are ideals of the quotient ring R[x]/(x(p)s-λ). In this paper, we derive necessary and sufficient conditions under which the quotient ring R[x]/(x(p)s- λ) is a chain ring. When R[x]/(x(p)s- λ) is a chain ring, all λ-constacyclic codes of length ps over R are known. In this paper, we establish the algebraic structures of all λ-constacyclic codes of length ps over R when R[x]/(x(p)s- λ) is a non-chain ring. We also determine the number of codewords in each of these codes. Using their algebraic structures, we obtain symbol-pair distances, Rosenbloom-Tsfasman (RT) distances, and RT weight distributions of all constacyclic codes of length ps over R. Apart from this, we derive necessary and sufficient conditions under which a constacyclic code of length ps over R is maximumdistance separable with respect to the: 1) Hamming metric; 2) symbol-pair metric; and 3) RT metric. We also provide an algorithm to decode the constacyclic codes of length ps over R using the known decoding algorithms of linear codes over finite fields with respect to the Hamming, symbol-pair, and RT metrics.

Let$p$be a prime,$s$be a positive integer, and let$b$be an integer satisfying$2 \leq b < p^{s}$. In this paper, we obtain$b$-symbol distances of all repeated-root constacyclic codes of length$p^{s}$over finite fields. Using this result, we determine$b$-symbol distances of all repeated-root constacyclic codes of length$p^{s}$over finite commutative chain rings. We also list all MDS$b$-symbol repeated-root constacyclic codes of length$p^{s}$over finite fields, and all MDS$b$-symbol repeated-root constacyclic codes of length$p^{s}$over finite commutative chain rings in general.