American Institute of Mathematical Sciences

Advanced
February  2009, 3(1): 53-58. doi: 10.3934/amc.2009.3.53

Finding an asymptotically bad family of $q$-th power residue codes

P. Charters 1,

1. 

The University of Texas at Austin, Mathematics Department, C1200, 1 University Station, Austin, TX, 78712, United States

Received  October 2008 Revised  January 2009 Published  January 2009

In coding theory, we are often interested in finding codes with a ''good'' relative minimum distance. To create a code that transmits information efficiently, we would like to see the ratio of bits containing information to total codeword length be large. In particular, for families of codes this means that as the codeword length increases, we do not significantly decrease the amount of information transmitted; for each fixed prime $q$ we can construct a family of $q$-ary codes with lengths $p$ and minimal distances $d_p$ with the property that as the length of these codes approaches infinity, the ratio of their minimal distance to their total length tends towards $\epsilon > 0$.
In this paper, we examine families of generalized binary quadratic residue codes, named $q$-th power residue codes, where $q$ is a fixed odd prime, and find an asymptotically bad subfamily of these codes. For each prime $l$ we will construct a $q$-th power residue code of length $p$ (determined by our choice of $l$ ) and minimal distance $d_p$, with the property that as $l$ approaches infinity, $p$ also tends towards infinity, and $\lim_{l to \infty} \frac{d_p}{p} = 0$.
Keywords: Chebotarev's density theorem., Coding theory, quadratic residue codes.
Mathematics Subject Classification: Primary: 11T71, 11H71; Secondary: 11R3.
Citation: P. Charters. Finding an asymptotically bad family of $q$-th power residue codes. Advances in Mathematics of Communications, 2009, 3 (1) : 53-58. doi: 10.3934/amc.2009.3.53
[1]

Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023
[2]

Min Ye, Alexander Barg. Polar codes for distributed hierarchical source coding. Advances in Mathematics of Communications, 2015, 9 (1) : 87-103. doi: 10.3934/amc.2015.9.87
[3]

Hai Q. Dinh, Hien D. T. Nguyen. On some classes of constacyclic codes over polynomial residue rings. Advances in Mathematics of Communications, 2012, 6 (2) : 175-191. doi: 10.3934/amc.2012.6.175
[4]

Karim Samei, Arezoo Soufi. Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$. Advances in Mathematics of Communications, 2017, 11 (4) : 791-804. doi: 10.3934/amc.2017058
[5]

Carla Mascia, Giancarlo Rinaldo, Massimiliano Sala. Hilbert quasi-polynomial for order domains and application to coding theory. Advances in Mathematics of Communications, 2018, 12 (2) : 287-301. doi: 10.3934/amc.2018018
[6]

Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035
[7]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367
[8]

Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011
[9]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374
[10]

Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313
[11]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109
[12]

Anish Ghosh, Dubi Kelmer. A quantitative Oppenheim theorem for generic ternary quadratic forms. Journal of Modern Dynamics, 2018, 12: 1-8. doi: 10.3934/jmd.2018001
[13]

Gokhan Calis, O. Ozan Koyluoglu. Architecture-aware coding for distributed storage: Repairable block failure resilient codes. Advances in Mathematics of Communications, 2018, 12 (3) : 465-503. doi: 10.3934/amc.2018028
[14]

Fabio Cipriani, Gabriele Grillo. On the $l^p$ -agmon's theory. Conference Publications, 1998, 1998 (Special) : 167-176. doi: 10.3934/proc.1998.1998.167
[15]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
[16]

V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413
[17]

Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555
[18]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945
[19]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[20]

Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences