# American Institute of Mathematical Sciences

February  2017, 11(1): 115-122. doi: 10.3934/amc.2017006

## Generalized Hamming weights of codes over the $\mathcal{GH}$ curve

 Universidade Federal de Uberlândia, Campus Santa Mônica, CEP 38.408-100, Av. Joao Naves de Avila, 2121, Uberlandia-MG, Brazil

Received  June 2015 Revised  February 2016 Published  February 2017

Fund Project: The author would like to thank FAPEMIG by support.

In this work, we studied the generalized Hamming weights of algebraic geometric Goppa codes on the $\mathcal{GH}$ curve. Especially, exact results on the second generalized Hamming weight are given for almost all cases. Furthermore, we apply the results obtained to show an example where the weight hierarchy characterizes the performance of the $\mathcal{GH}$ codes on a noiseless communication channel.

Citation: Alonso sepúlveda Castellanos. Generalized Hamming weights of codes over the $\mathcal{GH}$ curve. Advances in Mathematics of Communications, 2017, 11 (1) : 115-122. doi: 10.3934/amc.2017006
##### References:

show all references

##### References:
 [1] Olav Geil, Stefano Martin. Relative generalized Hamming weights of q-ary Reed-Muller codes. Advances in Mathematics of Communications, 2017, 11 (3) : 503-531. doi: 10.3934/amc.2017041 [2] 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 [3] Relinde Jurrius, Ruud Pellikaan. On defining generalized rank weights. Advances in Mathematics of Communications, 2017, 11 (1) : 225-235. doi: 10.3934/amc.2017014 [4] Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029 [5] Ming Su, Arne Winterhof. Hamming correlation of higher order. Advances in Mathematics of Communications, 2018, 12 (3) : 505-513. doi: 10.3934/amc.2018029 [6] Koray Karabina, Berkant Ustaoglu. Invalid-curve attacks on (hyper)elliptic curve cryptosystems. Advances in Mathematics of Communications, 2010, 4 (3) : 307-321. doi: 10.3934/amc.2010.4.307 [7] Robert L. Devaney, Daniel M. Look. Buried Sierpinski curve Julia sets. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1035-1046. doi: 10.3934/dcds.2005.13.1035 [8] Joaquim Borges, Josep Rifà, Victor Zinoviev. Completely regular codes by concatenating Hamming codes. Advances in Mathematics of Communications, 2018, 12 (2) : 337-349. doi: 10.3934/amc.2018021 [9] Gérard Cohen, Alexander Vardy. Duality between packings and coverings of the Hamming space. Advances in Mathematics of Communications, 2007, 1 (1) : 93-97. doi: 10.3934/amc.2007.1.93 [10] Yiwei Liu, Zihui Liu. On some classes of codes with a few weights. Advances in Mathematics of Communications, 2018, 12 (2) : 415-428. doi: 10.3934/amc.2018025 [11] Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321 [12] Emanuela Caliceti, Sandro Graffi. An existence criterion for the $\mathcal{PT}$-symmetric phase transition. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1955-1967. doi: 10.3934/dcdsb.2014.19.1955 [13] Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2469-2479. doi: 10.3934/dcds.2014.34.2469 [14] Diego F. Aranha, Ricardo Dahab, Julio López, Leonardo B. Oliveira. Efficient implementation of elliptic curve cryptography in wireless sensors. Advances in Mathematics of Communications, 2010, 4 (2) : 169-187. doi: 10.3934/amc.2010.4.169 [15] Huaiyu Jian, Hongjie Ju, Wei Sun. Traveling fronts of curve flow with external force field. Communications on Pure & Applied Analysis, 2010, 9 (4) : 975-986. doi: 10.3934/cpaa.2010.9.975 [16] Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063 [17] Marek Janasz, Piotr Pokora. On Seshadri constants and point-curve configurations. Electronic Research Archive, 2020, 28 (2) : 795-805. doi: 10.3934/era.2020040 [18] 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 [19] Olof Heden, Faina I. Solov’eva. Partitions of $\mathbb F$n into non-parallel Hamming codes. Advances in Mathematics of Communications, 2009, 3 (4) : 385-397. doi: 10.3934/amc.2009.3.385 [20] Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55

2019 Impact Factor: 0.734