-
Previous Article
Generalized nonlinearity of $ S$-boxes
- AMC Home
- This Issue
-
Next Article
Power decoding Reed-Solomon codes up to the Johnson radius
Singleton bounds for R-additive codes
Department of Mathematics, Bu Ali Sina University, Hamedan, Iran |
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.
References:
| [1] |
J. Bierbrauer,
The theory of cyclic codes and a generalization to additive codes, Des. Codes Crypt., 25 (2002), 189-206.
|
| [2] |
J. Bierbrauer,
Cyclic additive codes, J. Algebra, 372 (2012), 661-672.
|
| [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.
|
| [4] |
S. T. Dougherty and K. Shiromoto,
Maximum distance codes over rings of order 4, IEEE Trans. Inf. Theory, 47 (2001), 400-404.
|
| [5] |
T. Honold and I. Landjev,
Linear codes over finite chain rings, J. Combinatorics, 7 (2001), R11-R11.
|
| [6] |
W. C. Huffman,
Cyclic $ \mathbb F_q$-linear $ \mathbb F_{q^t}$-codes, Int. J. Inf. Coding Theory, 1 (2010), 249-284.
|
| [7] |
W. C. Huffman,
On the theory of $ \mathbb F_q$-linear $ \mathbb F_{q^t}$-codes, Adv. Math. Commun., 7 (2013), 349-378.
|
| [8] |
K. Samei and S. Mahmoudi,
Cyclic R-additive codes, Discrete Math., 340 (2017), 1657-1668.
|
| [9] |
K. Samei and S. Mahmoudi,
SR-additive codes, Submitted. |
| [10] |
K. Samei and A. Soufi, Constacyclic codes over finite principal ideals rings, Submitted. |
| [11] |
R. Y. Sharp,
Steps in Commutative Algebra, Cambridge Univ. Press, 1991. |
| [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.
|
| [13] |
J. Wood, Lecture notes on the MacWilliams identities and extension theorem, in Lect. Simat Int. School Conf. Coding Theory, 2008. |
show all references
References:
| [1] |
J. Bierbrauer,
The theory of cyclic codes and a generalization to additive codes, Des. Codes Crypt., 25 (2002), 189-206.
|
| [2] |
J. Bierbrauer,
Cyclic additive codes, J. Algebra, 372 (2012), 661-672.
|
| [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.
|
| [4] |
S. T. Dougherty and K. Shiromoto,
Maximum distance codes over rings of order 4, IEEE Trans. Inf. Theory, 47 (2001), 400-404.
|
| [5] |
T. Honold and I. Landjev,
Linear codes over finite chain rings, J. Combinatorics, 7 (2001), R11-R11.
|
| [6] |
W. C. Huffman,
Cyclic $ \mathbb F_q$-linear $ \mathbb F_{q^t}$-codes, Int. J. Inf. Coding Theory, 1 (2010), 249-284.
|
| [7] |
W. C. Huffman,
On the theory of $ \mathbb F_q$-linear $ \mathbb F_{q^t}$-codes, Adv. Math. Commun., 7 (2013), 349-378.
|
| [8] |
K. Samei and S. Mahmoudi,
Cyclic R-additive codes, Discrete Math., 340 (2017), 1657-1668.
|
| [9] |
K. Samei and S. Mahmoudi,
SR-additive codes, Submitted. |
| [10] |
K. Samei and A. Soufi, Constacyclic codes over finite principal ideals rings, Submitted. |
| [11] |
R. Y. Sharp,
Steps in Commutative Algebra, Cambridge Univ. Press, 1991. |
| [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.
|
| [13] |
J. Wood, Lecture notes on the MacWilliams identities and extension theorem, in Lect. Simat Int. School Conf. Coding Theory, 2008. |
| [1] |
Zihui Liu. Galois LCD codes over rings. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022002 |
| [2] |
Delphine Boucher, Patrick Solé, Felix Ulmer. Skew constacyclic codes over Galois rings. Advances in Mathematics of Communications, 2008, 2 (3) : 273-292. doi: 10.3934/amc.2008.2.273 |
| [3] |
Sergio R. López-Permouth, Steve Szabo. On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings. Advances in Mathematics of Communications, 2009, 3 (4) : 409-420. doi: 10.3934/amc.2009.3.409 |
| [4] |
Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011 |
| [5] |
David Grant, Mahesh K. Varanasi. The equivalence of space-time codes and codes defined over finite fields and Galois rings. Advances in Mathematics of Communications, 2008, 2 (2) : 131-145. doi: 10.3934/amc.2008.2.131 |
| [6] |
Ferruh Özbudak, Patrick Solé. Gilbert-Varshamov type bounds for linear codes over finite chain rings. Advances in Mathematics of Communications, 2007, 1 (1) : 99-109. doi: 10.3934/amc.2007.1.99 |
| [7] |
Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249 |
| [8] |
Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004 |
| [9] |
Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028 |
| [10] |
Aicha Batoul, Kenza Guenda, T. Aaron Gulliver. Some constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2016, 10 (4) : 683-694. doi: 10.3934/amc.2016034 |
| [11] |
Somphong Jitman, San Ling, Patanee Udomkavanich. Skew constacyclic codes over finite chain rings. Advances in Mathematics of Communications, 2012, 6 (1) : 39-63. doi: 10.3934/amc.2012.6.39 |
| [12] |
Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253 |
| [13] |
Eimear Byrne. On the weight distribution of codes over finite rings. Advances in Mathematics of Communications, 2011, 5 (2) : 395-406. doi: 10.3934/amc.2011.5.395 |
| [14] |
Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012 |
| [15] |
Gianira N. Alfarano, Anina Gruica, Julia Lieb, Joachim Rosenthal. Convolutional codes over finite chain rings, MDP codes and their characterization. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022028 |
| [16] |
Jean Creignou, Hervé Diet. Linear programming bounds for unitary codes. Advances in Mathematics of Communications, 2010, 4 (3) : 323-344. doi: 10.3934/amc.2010.4.323 |
| [17] |
Simeon Ball, Guillermo Gamboa, Michel Lavrauw. On additive MDS codes over small fields. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021024 |
| [18] |
Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035 |
| [19] |
Zihui Liu, Dajian Liao. Higher weights and near-MDR codes over chain rings. Advances in Mathematics of Communications, 2018, 12 (4) : 761-772. doi: 10.3934/amc.2018045 |
| [20] |
Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005 |
2021 Impact Factor: 1.015
Tools
Metrics
Other articles
by authors
[Back to Top]