154 research outputs found
Minimum distance of Symplectic Grassmann codes
We introduce the Symplectic Grassmann codes as projective codes defined by
symplectic Grassmannians, in analogy with the orthogonal Grassmann codes
introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special
class of Symplectic Grassmann codes. We describe the weight enumerator of the
Lagrangian--Grassmannian codes of rank and and we determine the minimum
distance of the line Symplectic Grassmann codes.Comment: Revised contents and biblograph
Applications of finite geometries to designs and codes
This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures.
A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples.
We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs.
Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes
An asymptotic property of quaternary additive codes
Let be the maximal length such that a quaternary additive
-code exists. We solve a natural asymptotic problem by determining
the lim sup of and the smallest value of such that
Our new family of quaternary additive codes has
parameters (where
and is an odd integer). These are constant-weight codes. The binary
codes obtained by concatenation meet the Griesmer bound with equality. The
proof is in terms of multisets of lines in Comment: 8 page
Exceptional scattered sequences
The concept of scattered polynomials is generalized to those of exceptional
scattered sequences which are shown to be the natural algebraic counterpart of
-linear MRD codes. The first infinite family in the first
nontrivial case is also provided and equivalence issues are considered. As a
byproduct, a new infinite family of MRD codes is obtained.Comment: 32 page
On subspace designs
Guruswami and Xing introduced subspace designs in 2013 to give the first
construction of positive rate rank metric codes list-decodable beyond half the
distance. In this paper we provide bounds involving the parameters of a
subspace design, showing they are tight via explicit constructions. We point
out a connection with sum-rank metric codes, dealing with optimal codes and
minimal codes with respect to this metric. Applications to two-intersection
sets with respect to hyperplanes, two-weight codes, cutting blocking sets and
lossless dimension expanders are also provided
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