The Extension Theorem with Respect to Symmetrized Weight Compositions

ElGarem, Noha; Megahed, Nefertiti

doi:10.1007/978-3-319-17296-5_18

Cited by 6 publications

(7 citation statements)

References 12 publications

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“…Regarding the symmetrized weight composition, the extension theorem for the case of classical linear codes was proved by Goldberg in [8]. There is a recent result in [7] where the authors proved that if an alphabet has a cyclic socle, then an analogue of the extension theorem holds for the symmetrized weight composition built on arbitrary group. The result was improved in [1] where the author showed, in some additional assumptions, the maximality of the cyclic socle condition for the symmetrized weight composition built on the full automorphism group of an alphabet.…”

Section: Introductionmentioning

confidence: 99%

When the extension property does not hold

Dyshko

2017

J. Algebra Appl.

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Section: Introductionmentioning

confidence: 99%

When the extension property does not hold

Dyshko

2017

J. Algebra Appl.

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“…It implies in particular that these maps are given by monomial matrices. The following result has been proven in a different manner by ElGarem et al [6,Thm. 13] and also appears in [22,Thm.…”

Section: Definition 43 a Weight Functionmentioning

confidence: 71%

“…and each side of the identity is a sum of elements in the character group C. Fix r ∈ R n and assume that r is contained in the block P of P R n , U T . Remark 2.1 (6) implies that the character χ( f (• ), r ) must appear on the right hand side of the above identity. In other words, there exists r ′ ∈ P i.e., r ′ = (A r r T ) T for some A r ∈ U , such that χ( f (• ), r ) = χ( • , r ′ ).…”

Section: Proof Let P Be a Block Ofmentioning

confidence: 99%

Extension theorems for various weight functions over Frobenius bimodules

Gluesing-Luerssen

Pllaha

2018

J. Algebra Appl.

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In this paper we study codes where the alphabet is a finite Frobenius bimodule over a finite ring. We discuss the extension property for various weight functions. Employing an entirely character-theoretic approach and a duality theory for partitions on Frobenius bimodules we derive alternative proofs for the facts that the Hamming weight and the homogeneous weight satisfy the extension property. We also use the same techniques to derive the extension property for other weights, such as the Rosenbloom-Tsfasman weight. . ⊥ K := {r ∈ R | rK = 0}, K ⊥ := {s ∈ S | Ks = 0}, I ⊥ := {v ∈ M | Iv = 0}, ⊥ J := {v ∈ M | vJ = 0}. (2.11) Clearly, ⊥ K, ⊥ J are submodules of R R and R M , respectively, while K ⊥ , I ⊥ are submodules of S S and M S , respectively.

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“…In [6], J.A.Wood proved that Frobenius rings do have the extension property with respect to swc. Later, in [4], it was shown that, more generally, a left R-module A has the extension property with respect to swc if it can be embedded in the character group R (given the standard module structure).…”

Section: Definition 13 (Extension Property)mentioning

confidence: 99%

On Symmetrized Weight Compositions

Assem

2014

Preprint

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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 R A to satisfy the EP with respect to swc built on any subgroup of Aut R (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.

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The Extension Theorem with Respect to Symmetrized Weight Compositions

Cited by 6 publications

References 12 publications

When the extension property does not hold

When the extension property does not hold

Extension theorems for various weight functions over Frobenius bimodules

On Symmetrized Weight Compositions

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