Name
Abstract | ||
|---|---|---|
Toric codes are evaluation codes obtained from an integral convex polytope $P \subset {\mathbb R}^n$ and finite field ${\mathbb F}_q$. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently in [V. Diaz, C. Guevara, and M. Vath, Proceedings of Simu Summer Institute, 2001], [J. Hansen, Appl. Algebra Engrg. Comm. Comput., 13 (2002), pp. 289-300; Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), Springer, Berlin, pp. 132-142], and [D. Joyner, Appl. Algebra Engrg. Comm. Comput., 15 (2004), pp. 63-79]. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon $P \subset {\mathbb R}^2$ by examining Minkowski sum decompositions of subpolygons of $P$. Our results give a simple and unifying explanation of bounds in Hansen’s work and empirical results of Joyner; they also apply to previously unknown cases. |
Year | DOI | Venue |
|---|---|---|
2006 | 10.1137/050637054 | SIAM J. Discrete Math. |
Keywords | Field | DocType |
minkowski sums,minkowski sum.,algebra engrg,reed-solomon code,mathbb r,j. hansen,toric code,toric surface codes,. toric variety,m. vath,coding theory,minkowski sum decomposition,mathbb f,c. guevara,algebraic geometry,convex polytope,toric variety,reed solomon code,finite field,minkowski sum,upper and lower bounds | Toric variety,Discrete mathematics,Finite field,Combinatorics,Upper and lower bounds,Toric code,Minkowski space,Convex polytope,Coding theory,Minkowski addition,Mathematics | Journal |
Volume | Issue | ISSN |
20 | 4 | SIAM Journal on Discrete Mathematics, 20, (2006), 999-1014 |
Citations | PageRank | References |
24 | 2.46 | 4 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| John Little | 1 | 24 | 2.80 |
| Hal Schenck | 2 | 30 | 4.50 |