Singleton bounds for R-additive codes

Karim Samei; Saadoun Mahmoudi

doi:10.3934/amc.2018006

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2018, Volume 12, Issue 1: 107-114. Doi: 10.3934/amc.2018006

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Singleton bounds for R-additive codes

Karim Samei and
Saadoun Mahmoudi

Department of Mathematics, Bu Ali Sina University, Hamedan, Iran

Received: June 30, 2016

Revised: March 31, 2017

Published: February 2018

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Abstract

Shiromoto (Linear Algebra Applic 295 (1999) 191-200) obtained the basic exact sequence for the Lee and Euclidean weights of linear codes over $ \mathbb{Z}_{\ell}$ and as an application, he found the Singleton bounds for linear codes over $ \mathbb{Z}_{\ell}$ with respect to Lee and Euclidean weights. Huffman (Adv. Math. Commun 7 (3) (2013) 349-378) obtained the Singleton bound for $ \mathbb{F}_{q}$-linear $ \mathbb{F}_{q^{t}}$-codes with respect to Hamming weight. Recently the theory of $ \mathbb{F}_{q}$-linear $ \mathbb{F}_{q^{t}}$-codes were generalized to $ R$-additive codes over $ R$-algebras by Samei and Mahmoudi. In this paper, we generalize Shiromoto's results for linear codes over $ \mathbb{Z}_{\ell}$ to $ R$-additive codes. As an application, when $ R$ is a chain ring, we obtain the Singleton bounds for $ R$-additive codes over free $ R$-algebras. Among other results, the Singleton bounds for additive codes over Galois rings are given.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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References

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