$ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $

Pascal-Brianchon Sets in Pappian Projective Planes Pascal-Brianchon Sets in Pappian Projective Planes

It is well-known that Pascal and Brianchon theorems characterize conics in a Pappian projective plane. But, using these theorems and their modifications we shall show that the notion of a conic (or better a Pascal-Brianchon set) can be defined without any use of theory of projectivities or of polarities as usually.

A new construction of self-dual codes from projective planes

Special Point Sets in Finite Projective Planes

We consider the following three problems: 1. Let U be a q-subset of GF(q 2) with the properties 0; 1 2 U and u ? v is a square for all u; v 2 U. Does it follow that U consists of the elements of the subbeld GF(q)? Here q is odd. 2. Let f : GF(q) ! GF(q) be any function, and let be the set of diierence quotients (directions, slopes). What are the possibilities for jD f j? 3. Let B be a subset of...

Resolving Sets and Semi-Resolving Sets in Finite Projective Planes

In a graph Γ = (V,E) a vertex v is resolved by a vertex-set S = {v1, . . . , vn} if its (ordered) distance list with respect to S, (d(v, v1), . . . , d(v, vn)), is unique. A set A ⊂ V is resolved by S if all its elements are resolved by S. S is a resolving set in Γ if it resolves V . The metric dimension of Γ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving...

PD-sets for Z4-linear codes: Hadamard and Kerdock codes

Permutation decoding is a technique that strongly depends on the existence of a special subset, called PD-set, of the permutation automorphism group of a code. In this paper, a general criterion to obtain s-PD-sets of size s + 1, which enable correction up to s errors, for Z4-linear codes is provided. Furthermore, some explicit constructions of s-PD-sets of size s+1 for important families of (n...