On Symmetrized Weight Compositions

A characterization of module alphabets with the Hamming weight EP (abbreviation for Extension Property) had been settled. A thoughtfully constructed piece-of-art example by J.A.Wood ([7]) finished the tour. In 2009, in [8], Frobenius bimodules were proved to satisfy the EP with respect to symmetrized weight compositions. In [4], the embeddability in the character group of the ambient ring R was found sufficient for a module RA to satisfy the EP with respect to swc built on any subgroup of AutR(A), while the necessity remained a question. Here, landing in a “Midway”, the necessity is proved by jumping to Hamming weight. Corollary 1.11 declares a characterization of module alphabets satisfying the EP with respect to swc. Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for functions is that inputs are to the left. Symmetrized Weight Compositions Definition 1.1. (Symmetrized Weight Compositions) Let G be a subgroup of the automorphism group of a finite R-module A. Define ∼ on A by a ∼ b if a = bτ for some τ ∈ G. Let A/G denote the orbit space of this action. The symmetrized weight composition is a function swc : A ×A/G → Q defined by, swc(x, a) = swca(x) = |{i : xi ∼ a}|, where x = (x1, . . . , xn) ∈ A and a ∈ A/G. Definition 1.2. Let G be a subgroup of AutR(A), a map T ia called a G-monomial transformation of A if for any (x1, . . . , xn) ∈ A (x1, . . . , xn)T = (xσ(1)τ1, . . . , xσ(n)τn), where σ ∈ Sn and τi ∈ G for i = 1, . . . , n. Definition 1.3. (Extension Property) The alphabet A has the extension property with respect to swc if for every n, and any two linear codes C1, C2 ⊂ A, any R-linear isomorphism f : C1 → C2 that preserves swc is extendable to a G-monomial transformation of A.