# American Institute of Mathematical Sciences

November  2009, 3(4): 409-420. doi: 10.3934/amc.2009.3.409

## On the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings

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

Received  July 2009 Revised  September 2009 Published  November 2009

Repeated root Cyclic and Negacyclic codes over Galois rings have been studied much less than their simple root counterparts. This situation is beginning to change. For example, repeated root codes of length ps, where $p$ is the characteristic of the alphabet ring, have been studied under some additional hypotheses. In each one of those cases, the ambient space for the codes has turned out to be a chain ring. In this paper, all remaining cases of cyclic and negacyclic codes of length ps over a Galois ring alphabet are considered. In these cases the ambient space is a local ring with simple socle but not a chain ring. Nonetheless, by reducing the problem to one dealing with uniserial subambients, a method for computing the Hamming distance of these codes is provided.
Citation: 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
 [1] San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038 [2] Saadoun Mahmoudi, Karim Samei. Codes over $\frak m$-adic completion rings. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020122 [3] Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $2^kn$ over $\mathbb{Z}_4+u\mathbb{Z}_4$. Advances in Mathematics of Communications, 2021, 15 (2) : 291-309. doi: 10.3934/amc.2020067 [4] Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020124 [5] Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $A_n$-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118 [6] Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020129 [7] Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044 [8] Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045 [9] Karan Khathuria, Joachim Rosenthal, Violetta Weger. Encryption scheme based on expanded Reed-Solomon codes. Advances in Mathematics of Communications, 2021, 15 (2) : 207-218. doi: 10.3934/amc.2020053 [10] Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065 [11] Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055 [12] Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054 [13] Shanding Xu, Longjiang Qu, Xiwang Cao. Three classes of partitioned difference families and their optimal constant composition codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020120 [14] Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049 [15] Hongming Ru, Chunming Tang, Yanfeng Qi, Yuxiao Deng. A construction of $p$-ary linear codes with two or three weights. Advances in Mathematics of Communications, 2021, 15 (1) : 9-22. doi: 10.3934/amc.2020039 [16] Tingting Wu, Li Liu, Lanqiang Li, Shixin Zhu. Repeated-root constacyclic codes of length $6lp^s$. Advances in Mathematics of Communications, 2021, 15 (1) : 167-189. doi: 10.3934/amc.2020051 [17] Hongwei Liu, Jingge Liu. On $\sigma$-self-orthogonal constacyclic codes over $\mathbb F_{p^m}+u\mathbb F_{p^m}$. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127 [18] Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $C_{2t}\times C_2$. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041 [19] Akbar Mahmoodi Rishakani, Seyed Mojtaba Dehnavi, Mohmmadreza Mirzaee Shamsabad, Nasour Bagheri. Cryptographic properties of cyclic binary matrices. Advances in Mathematics of Communications, 2021, 15 (2) : 311-327. doi: 10.3934/amc.2020068 [20] Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $8p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020123

2019 Impact Factor: 0.734