arXiv subjects

David Joyner

Publications and source records attributed to David Joyner.

21 records · Page 2Linked to original sources

arXiv2003

Toric codes over finite fields

In this note, a class of error-correcting codes is associated to a toric variety associated to a fan defined over a finite field $\fff_q$, analogous to the class of Goppa codes associated to a curve. For such a ``toric code'' satisfying certain additional conditions, we present an efficient decoding algorithm for the dual of a Goppa code. Many examples are given. For small $q$, many of these codes have parameters beating the Gilbert-Varshamov bound. In fact, using toric codes, we construct a $(n,k,d)=(49,11,28)$ code over $\fff_8$, which is better than any other known code listed in Brouwer's on-line tables for that $n$ and $k$.

math.AG↗

arXiv2003

On the variety of Borels in relative position $\vec{w}$

Let $G$ be a connected semi-simple group defined over and algebraically closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set of simple reflections associated to the simple positive roots corresponding to $(T,B)$, and let ${\cal B}\cong G/B$ denote the Borel variety. For any $s_i\in S$, $1\leq i\leq n$, let $\bar{O}(s_1,..., s_n)= \{(B_0,..., B_{n})\in {\cal B}^{n+1} | (B_{i-1},B_{i})\in \bar{O(s_i)}, 1\leq i\leq n\}$, where $O(s)$ denotes the subvariety of pairs of Borels in ${\cal B}^2$ in relative position $s$. We show that such varieties are smooth and indicate why this result is, in one sense, best possible. Our main results assume that $k$ has characteristic 0.

math.AG↗

arXiv2002

Notes on toric varieties

These notes survey some basic results in toric varieties over a field with examples and applications. A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any number of dimensions (written in both the software packages MAGMA and GAP). Among other things, the package implements a desingularization procedure for affine toric varieties, constructs some error-correcting codes associated with toric varieties, and computes the Riemann-Roch space of a divisor on a toric variety.

math.AG↗