American Institute of Mathematical Sciences

August  2009, 3(3): 227-234. doi: 10.3934/amc.2009.3.227

Dual generalizations of the concept of cyclicity of codes

 1 Department of Mathematics, Ohio University, Athens, Ohio-45701, United States, United States, United States

Received  December 2008 Revised  May 2009 Published  August 2009

In this paper we focus on two generalizations of the notion of cyclicity of codes: polycyclic codes and sequential codes. We establish a duality between these two generalizations and also show connections between them and other well-known generalizations of cyclicity such as the notions of negacyclicity and constacyclicity. In particular, it is shown that a code $C$ is sequential and polycyclic if and only if $C$ and its dual C are both sequential if and only if $C$ and its dual C are both polycyclic. Furthermore, any one of these equivalent statements characterizes the family of constacyclic codes.
Citation: Sergio R. López-Permouth, Benigno R. Parra-Avila, Steve Szabo. Dual generalizations of the concept of cyclicity of codes. Advances in Mathematics of Communications, 2009, 3 (3) : 227-234. doi: 10.3934/amc.2009.3.227
 [1] Adel Alahmadi, Steven Dougherty, André Leroy, Patrick Solé. On the duality and the direction of polycyclic codes. Advances in Mathematics of Communications, 2016, 10 (4) : 921-929. doi: 10.3934/amc.2016049 [2] Alexis Eduardo Almendras Valdebenito, Andrea Luigi Tironi. On the dual codes of skew constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 659-679. doi: 10.3934/amc.2018039 [3] Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039 [4] 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 [5] Ram Krishna Verma, Om Prakash, Ashutosh Singh, Habibul Islam. New quantum codes from skew constacyclic codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021028 [6] Ranya Djihad Boulanouar, Aicha Batoul, Delphine Boucher. An overview on skew constacyclic codes and their subclass of LCD codes. Advances in Mathematics of Communications, 2021, 15 (4) : 611-632. doi: 10.3934/amc.2020085 [7] Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543 [8] Alexandre Fotue-Tabue, Edgar Martínez-Moro, J. Thomas Blackford. On polycyclic codes over a finite chain ring. Advances in Mathematics of Communications, 2020, 14 (3) : 455-466. doi: 10.3934/amc.2020028 [9] Jinmei Fan, Yanhai Zhang. Optimal quinary negacyclic codes with minimum distance four. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021043 [10] 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 [11] 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 [12] Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177 [13] Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 [14] Gustavo Terra Bastos, Reginaldo Palazzo Júnior, Marinês Guerreiro. Abelian non-cyclic orbit codes and multishot subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 631-650. doi: 10.3934/amc.2020035 [15] Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41 [16] Yunwen Liu, Longjiang Qu, Chao Li. New constructions of systematic authentication codes from three classes of cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 1-16. doi: 10.3934/amc.2018001 [17] Nuh Aydin, Nicholas Connolly, Markus Grassl. Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes. Advances in Mathematics of Communications, 2017, 11 (1) : 245-258. doi: 10.3934/amc.2017016 [18] 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 [19] 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 [20] Anuradha Sharma, Saroj Rani. Trace description and Hamming weights of irreducible constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 123-141. doi: 10.3934/amc.2018008

2020 Impact Factor: 0.935