Some remarks on non projective Frobenius algebras and linear codes

Abstract

With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over their characteristic subrings. From the perspective of linear codes, our work expands one's options to construct new finite Frobenius rings from old ones. We close with a discussion of generalized versions of the MacWilliams identities that may be obtained in this context.

Decoding Linear Codes over Chain Rings Given by Parity Check Matrices

“…as a consequence of a recursive application of the identities π i [i] = id m i /m i+1 , (4) and (3). Finally, we may see that…”

“…For instance, Wood, in [2], states MacWilliams identities for finite Frobenius rings, extending the foundations of coding theory to linear codes over Frobenius rings. In [3] it is proven that finite Frobenius rings are Frobenius algebras over their characteristic subrings, which enriches the duality theory for linear codes over this kind of alphabet.…”

Linear codes over finite rings are trace codes

“…3 Trace functions on finite rings To avoid misunderstandings, we denote such a trace also by the symbol Tr S R , whereas we omit this addition, whenever such a misunderstanding is excluded. A careful treatment of ring extensions and according trace functions can be found in the recent paper [9] by J. G.-Torrecillas et al…”

Noncommutative Coding Theory and Algebraic Sets for Skew PBW Extensions

“…Applying the classical Hilbert's Nullstellensatz for Z(A) (here we use that F is algeraically closed) we have √ J = I Z(A) (V Z(A) (J)), so we get that I Z(A) (V Z(A) (J)) ⊆ √ I. (ii) If q = 1 is an arbitrary root of unity of degree m ≥ 2, then the center of the quantum plane A := F q [x, y] is the subalgebra generated by x m and y m , i.e., Z(F q [x, y]) = F[x m , y m ] (see [37] or also [32], Proposition 3.3.14). Recall that the rule of multiplication in A is given by yx = qxy.…”