# American Institute of Mathematical Sciences

February  2018, 12(1): 107-114. doi: 10.3934/amc.2018006

## Singleton bounds for R-additive codes

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

Received  July 2016 Revised  April 2017 Published  March 2018

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.

Citation: Karim Samei, Saadoun Mahmoudi. Singleton bounds for R-additive codes. Advances in Mathematics of Communications, 2018, 12 (1) : 107-114. doi: 10.3934/amc.2018006
##### References:
 [1] J. Bierbrauer, The theory of cyclic codes and a generalization to additive codes, Des. Codes Crypt., 25 (2002), 189-206.   Google Scholar [2] J. Bierbrauer, Cyclic additive codes, J. Algebra, 372 (2012), 661-672.   Google Scholar [3] B. K. Dey and B. S. Rajan, $\mathbb F_q$-linear cyclic codes over $\mathbb F_{q^m}$: DFT approach, Des. Codes Crypt., 34 (2005), 89-116.   Google Scholar [4] S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE Trans. Inf. Theory, 47 (2001), 400-404.   Google Scholar [5] T. Honold and I. Landjev, Linear codes over finite chain rings, J. Combinatorics, 7 (2001), R11-R11.   Google Scholar [6] W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Int. J. Inf. Coding Theory, 1 (2010), 249-284.   Google Scholar [7] W. C. Huffman, On the theory of $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Adv. Math. Commun., 7 (2013), 349-378.   Google Scholar [8] K. Samei and S. Mahmoudi, Cyclic R-additive codes, Discrete Math., 340 (2017), 1657-1668.   Google Scholar [9] K. Samei and S. Mahmoudi, SR-additive codes, Submitted. Google Scholar [10] K. Samei and A. Soufi, Constacyclic codes over finite principal ideals rings, Submitted. Google Scholar [11] R. Y. Sharp, Steps in Commutative Algebra, Cambridge Univ. Press, 1991.  Google Scholar [12] K. Shiromoto, A basic exact sequence for the Lee and Euclidean weights of linear codes over $\mathbb Z_{\ell}$, Linear Algebra Appl., 295 (1999), 191-200.   Google Scholar [13] J. Wood, Lecture notes on the MacWilliams identities and extension theorem, in Lect. Simat Int. School Conf. Coding Theory, 2008.  Google Scholar

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##### References:
 [1] J. Bierbrauer, The theory of cyclic codes and a generalization to additive codes, Des. Codes Crypt., 25 (2002), 189-206.   Google Scholar [2] J. Bierbrauer, Cyclic additive codes, J. Algebra, 372 (2012), 661-672.   Google Scholar [3] B. K. Dey and B. S. Rajan, $\mathbb F_q$-linear cyclic codes over $\mathbb F_{q^m}$: DFT approach, Des. Codes Crypt., 34 (2005), 89-116.   Google Scholar [4] S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE Trans. Inf. Theory, 47 (2001), 400-404.   Google Scholar [5] T. Honold and I. Landjev, Linear codes over finite chain rings, J. Combinatorics, 7 (2001), R11-R11.   Google Scholar [6] W. C. Huffman, Cyclic $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Int. J. Inf. Coding Theory, 1 (2010), 249-284.   Google Scholar [7] W. C. Huffman, On the theory of $\mathbb F_q$-linear $\mathbb F_{q^t}$-codes, Adv. Math. Commun., 7 (2013), 349-378.   Google Scholar [8] K. Samei and S. Mahmoudi, Cyclic R-additive codes, Discrete Math., 340 (2017), 1657-1668.   Google Scholar [9] K. Samei and S. Mahmoudi, SR-additive codes, Submitted. Google Scholar [10] K. Samei and A. Soufi, Constacyclic codes over finite principal ideals rings, Submitted. Google Scholar [11] R. Y. Sharp, Steps in Commutative Algebra, Cambridge Univ. Press, 1991.  Google Scholar [12] K. Shiromoto, A basic exact sequence for the Lee and Euclidean weights of linear codes over $\mathbb Z_{\ell}$, Linear Algebra Appl., 295 (1999), 191-200.   Google Scholar [13] J. Wood, Lecture notes on the MacWilliams identities and extension theorem, in Lect. Simat Int. School Conf. Coding Theory, 2008.  Google Scholar
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