25 research outputs found
Error Correcting Codes on Algebraic Surfaces
Error correcting codes are defined and important parameters for a code are
explained. Parameters of new codes constructed on algebraic surfaces are
studied. In particular, codes resulting from blowing up points in \proj^2 are
briefly studied, then codes resulting from ruled surfaces are covered. Codes
resulting from ruled surfaces over curves of genus 0 are completely analyzed,
and some codes are discovered that are better than direct product Reed Solomon
codes of similar length. Ruled surfaces over genus 1 curves are also studied,
but not all classes are completely analyzed. However, in this case a family of
codes are found that are comparable in performance to the direct product code
of a Reed Solomon code and a Goppa code. Some further work is done on surfaces
from higher genus curves, but there remains much work to be done in this
direction to understand fully the resulting codes. Codes resulting from blowing
points on surfaces are also studied, obtaining necessary parameters for
constructing infinite families of such codes.
Also included is a paper giving explicit formulas for curves with more
\field{q}-rational points than were previously known for certain combinations
of field size and genus. Some upper bounds are now known to be optimal from
these examples.Comment: This is Chris Lomont's PhD thesis about error correcting codes from
algebriac surface
Balanced Product Quantum Codes
This work provides the first explicit and non-random family of
LDPC quantum codes which encode logical qubits
with distance . The family is constructed by
amalgamating classical codes and Ramanujan graphs via an operation called
balanced product.
Recently, Hastings-Haah-O'Donnell and Panteleev-Kalachev were the first to
show that there exist families of LDPC quantum codes which break the
distance barrier. However, their
constructions are based on probabilistic arguments which only guarantee the
code parameters with high probability whereas our bounds hold unconditionally.
Further, balanced products allow for non-abelian twisting of the check
matrices, leading to a construction of LDPC quantum codes that can be shown to
have and that we conjecture to have linear distance .Comment: 23 pages, 11 figure
Bounds on codes from smooth toric threefolds with rank(pic(x)) = 2
In 1998, J. P. Hansen introduced the construction of an error-correcting code over a
finite field Fq from a convex integral polytope in R2. Given a polytope P ⊂ R2, there
is an associated toric variety XP , and Hansen used the cohomology and intersection
theory of divisors on XP to determine explicit formulas for the dimension and minimum
distance of the associated toric code CP . We begin by reviewing the basics
of algebraic coding theory and toric varieties and discuss how these areas intertwine
with discrete geometry. Our first results characterize certain polygons that generate
and do not generate maximum distance separable (MDS) codes and Almost-MDS
codes. In 2006, Little and Schenck gave formulas for the minimum distance of certain
toric codes corresponding to smooth toric surfaces with rank(Pic(X)) = 2 and
rank(Pic(X)) = 3. Additionally, they gave upper and lower bounds on the minimum
distance of an arbitrary toric code CP by finding a subpolygon of P with a maximal,
nontrivial Minkowski sum decomposition. Following this example, we give explicit
formulas for the minimum distance of toric codes associated with two families of
smooth toric threefolds with rank(Pic(X)) = 2, characterized by G. Ewald and A.
Schmeinck in 1993. Lastly, we give explicit formulas for the dimension of a toric code
generated from a Minkowski sum of a finite number of polytopes in R2 and R3 and a
lower bound for the minimum distance