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Article Contents

# Singleton bounds for R-additive codes

• 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.

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

 Citation:

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