Properties of dual codes defined by nondegenerate forms

Szabo, Steve

doi:10.13069/jacodesmath.284934

Cited by 5 publications

(14 citation statements)

References 8 publications

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“…One of these equivalent conditions is the existence of a Frobenius functional, which will be seen to generalize the generating character and play a similar role to it. In Section 4, we extend results from [11] on annihilators associated to a non degenerate bilinear form from a finite Frobenius ring to a non projective Frobenius algebra. Section 5 contains our observation that, given an algebra R over a Frobenius commutative ring K such that R is finitely generated as a K-module, then R is a non projective Frobenius algebra over K if and only if R is a Frobenius ring.…”

Section: Introductionmentioning

confidence: 94%

“…(2) Isomorphisms of right A-modules α : A → A. Let us see that this framework covers the module-theoretical setting considered in [11].…”

Section: Codes With a Frobenius Alphabetmentioning

confidence: 99%

“…Let ǫ : A → Z n be a Frobenius functional (that is, a generating character). According to examples 24 and 25, this setting covers all cases considered in [11].…”

Section: Codes With a Frobenius Alphabetmentioning

confidence: 99%

“…A general version of MacWilliams identity appears in [11,Theorem 5.2]. Unfortunately, it is not valid for every non degenerate bilinear form, as the following example shows.…”

Section: Codes With a Frobenius Alphabetmentioning

confidence: 99%

“…From the point of view of codes, this gives a way to construct new finite Frobenius rings from old ones; this is further discussed in Section 6. That section also contains a discussion of how the general results on bilinear forms defined on modules over non projective Frobenius algebras, developed in the previous sections, may be applied to get the main results of [11]. We close that section with a discussion of those bilinear forms for which a version of MacWilliams identities stated in [11] holds.…”

Section: Introductionmentioning

confidence: 96%

See 4 more Smart Citations

Some remarks on non projective Frobenius algebras and linear codes

Gómez-Torrecillas

Hieta-aho

Lobillo

et al. 2019

Des. Codes Cryptogr.

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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.

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Section: Introductionmentioning

confidence: 94%

“…(2) Isomorphisms of right A-modules α : A → A. Let us see that this framework covers the module-theoretical setting considered in [11].…”

Section: Codes With a Frobenius Alphabetmentioning

confidence: 99%

“…Let ǫ : A → Z n be a Frobenius functional (that is, a generating character). According to examples 24 and 25, this setting covers all cases considered in [11].…”

Section: Codes With a Frobenius Alphabetmentioning

confidence: 99%

“…A general version of MacWilliams identity appears in [11,Theorem 5.2]. Unfortunately, it is not valid for every non degenerate bilinear form, as the following example shows.…”

Section: Codes With a Frobenius Alphabetmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 3 more Smart Citations

Some remarks on non projective Frobenius algebras and linear codes

Gómez-Torrecillas

Hieta-aho

Lobillo

et al. 2019

Des. Codes Cryptogr.

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On the structure of repeated-root polycyclic codes over local rings

Bajalan,

Martínez-Moro,

Sobhani

et al. 2024

Discrete Mathematics

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ℤ₂ℤ₄-additive codes as codes over rings

Fernández-Córdoba¹,

Szabo²

2023

Contemporary Mathematics

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In this paper, it is shown that Z 2 Z 4 \mathbb {Z}_2\mathbb {Z}_4 -additive codes with additional structure can be viewed as linear codes over rings. As an example, codes over the finite chain ring of order 8, R = Z 4 [ x ] ⟨ x 2 − 2 , 2 x ⟩ R=\frac {\mathbb {Z}_4[x]}{\langle x^2-2,2x\rangle } , which as additive group is isomorphic to Z 2 × Z 4 \mathbb {Z}_2\times \mathbb {Z}_4 , are shown to be Z 2 Z 4 \mathbb {Z}_2\mathbb {Z}_4 -additive codes. Amongst other results connecting linear codes over R R and their Z 2 Z 4 \mathbb {Z}_2\mathbb {Z}_4 -additive images, it is shown that the Z 2 Z 4 \mathbb {Z}_2\mathbb {Z}_4 -additive image of a cyclic code over R R is separable, that is, a direct sum of a binary linear code and a linear code over Z 4 \mathbb {Z}_4 . The family of chain rings, Z 4 [ x ] ⟨ x s − 2 , 2 x t ⟩ {\mathbb {Z}_4[x]\over \langle x^s-2, 2x^t\rangle } , where 1 ≤ t > s 1\leq t>s , and the finite commutative local Frobenius non-chain ring Z 4 [ x , y ] ⟨ x 2 − 2 , x y − 2 , y 2 , 2 x , 2 y ⟩ \frac {\mathbb {Z}_4[x,y]}{\langle x^2-2,xy-2,y^2,2x,2y\rangle } are also considered as alphabets for the study of Z 2 Z 4 \mathbb {Z}_2\mathbb {Z}_4 -additive codes.

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Properties of dual codes defined by nondegenerate forms

Cited by 5 publications

References 8 publications

Some remarks on non projective Frobenius algebras and linear codes

Some remarks on non projective Frobenius algebras and linear codes

On the structure of repeated-root polycyclic codes over local rings

ℤ₂ℤ₄-additive codes as codes over rings

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