Small polygons and toric codes
نویسندگان
چکیده
We describe two different approaches to making systematic classifications of plane lattice polygons, and recover the toric codes they generate, over small fields, where these match or exceed the best known minimum distance. This includes a [36, 19, 12]-code over F7 whose minimum distance 12 exceeds that of all previously known codes.
منابع مشابه
On classification of toric surface codes of low dimension
This work is a natural continuation of our previous work [YZ]. In this paper, we give a complete classification of toric surface codes of dimension equal to 6, except a special pair, C P (4) 6 and C P (5) 6 over F8. Also, we give an example, C P (5) 6 and C P (6) 6 over F7, to illustrate that two monomially equivalent toric codes can be constructed from two lattice non-equivalent polygons.
متن کاملOn dual toric complete intersection codes
In this paper we study duality for evaluation codes on intersections of d hypersurfaces with given d -dimensional Newton polytopes, so called toric complete intersection codes. In particular, we give a condition for such a code to be quasi-self-dual. In the case of d = 2 it reduces to a combinatorial condition on the Newton polygons. This allows us to give an explicit construction of dual and q...
متن کاملToric Surface Codes and Minkowski Length of Polygons
In this paper we prove new lower bounds for the minimum distance of a toric surface code CP defined by a convex lattice polygon P ⊂ R 2 . The bounds involve a geometric invariant L(P ) , called the full Minkowski length of P which can be easily computed for any given P .
متن کاملOn m-dimensional toric codes
Toric codes are a class of m-dimensional cyclic codes introduced recently by J. Hansen in [7], [8], and studied in [9], [5], [10]. They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope P ⊆ Rm. As such, they are in a sense a natural extension of Reed-Solomon codes. Several articles cited above use intersection theor...
متن کاملThe toric geometry of triangulated polygons in Euclidean space
Speyer and Sturmfels [SpSt] associated Gröbner toric degenerations Gr2(C) of Gr2(Cn) to each trivalent tree T with n leaves. These degenerations induce toric degenerations M r of Mr, the space of n ordered, weighted (by r) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers as stratified symplectic spaces and describe ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Symb. Comput.
دوره 51 شماره
صفحات -
تاریخ انتشار 2013