112 research outputs found
The use of blocking sets in Galois geometries and in related research areas
Blocking sets play a central role in Galois geometries. Besides their intrinsic geometrical importance, the importance of blocking sets also arises from the use of blocking sets for the solution of many other geometrical problems, and problems in related research areas. This article focusses on these applications to motivate researchers to investigate blocking sets, and to motivate researchers to investigate the problems that can be solved by using blocking sets. By showing the many applications on blocking sets, we also wish to prove that researchers who improve results on blocking sets in fact open the door to improvements on the solution of many other problems
A study of intersections of quadrics having applications on the small weight codewords of the functional codes C2(Q), Q a non-singular quadric
AbstractWe study the small weight codewords of the functional code C2(Q), with Q a non-singular quadric in PG(N,q). We prove that the small weight codewords correspond to the intersections of Q with the singular quadrics of PG(N,q) consisting of two hyperplanes. We also calculate the number of codewords having these small weights
Large weight code words in projective space codes
AbstractRecently, a large number of results have appeared on the small weights of the (dual) linear codes arising from finite projective spaces. We now focus on the large weights of these linear codes. For q even, this study for the code Ck(n,q)⊥ reduces to the theory of minimal blocking sets with respect to the k-spaces of PG(n,q), odd-blocking the k-spaces. For q odd, in a lot of cases, the maximum weight of the code Ck(n,q)⊥ is equal to qn+⋯+q+1, but some unexpected exceptions arise to this result. In particular, the maximum weight of the code C1(n,3)⊥ turns out to be 3n+3n-1. In general, the problem of whether the maximum weight of the code Ck(n,q)⊥, with q=3h (h⩾1), is equal to qn+⋯+q+1, reduces to the problem of the existence of sets of points in PG(n,q) intersecting every k-space in 2(mod3) points
On the minimum number of minimal codewords
We study the minimum number of minimal codewords in linear codes from the
point of view of projective geometry. We derive bounds and in some cases
determine the exact values. We also present an extension to minimal subcode
supports.Comment: 8 pages, 1 tabl
Small weight codewords of projective geometric codes II
The -ary linear code is defined as the row space of
the incidence matrix of -spaces and points of . It is
known that if is square, a codeword of weight exists that cannot be written as a linear combination
of at most rows of . Over the past few decades, researchers have
put a lot of effort towards proving that any codeword of smaller weight does
meet this property. We show that if is a composite prime
power, every codeword of up to weight is a linear combination of at most rows of
. We also generalise this result to the codes ,
which are defined as the -ary row span of the incidence matrix of -spaces
and -spaces, .Comment: 22 page
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