84 research outputs found
Enumerative Coding for Grassmannian Space
The Grassmannian space \Gr is the set of all dimensional subspaces of
the vector space~\smash{\F_q^n}. Recently, codes in the Grassmannian have
found an application in network coding. The main goal of this paper is to
present efficient enumerative encoding and decoding techniques for the
Grassmannian. These coding techniques are based on two different orders for the
Grassmannian induced by different representations of -dimensional subspaces
of \F_q^n. One enumerative coding method is based on a Ferrers diagram
representation and on an order for \Gr based on this representation. The
complexity of this enumerative coding is digit
operations. Another order of the Grassmannian is based on a combination of an
identifying vector and a reduced row echelon form representation of subspaces.
The complexity of the enumerative coding, based on this order, is
digits operations. A combination of the two
methods reduces the complexity on average by a constant factor.Comment: to appear in IEEE Transactions on Information Theor
Enumerative Encoding in the Grassmannian Space
Codes in the Grassmannian space have found recently application in network
coding. Representation of -dimensional subspaces of \F_q^n has generally
an essential role in solving coding problems in the Grassmannian, and in
particular in encoding subspaces of the Grassmannian. Different representations
of subspaces in the Grassmannian are presented. We use two of these
representations for enumerative encoding of the Grassmannian. One enumerative
encoding is based on Ferrers diagrams representation of subspaces; and another
is based on identifying vector and reduced row echelon form representation of
subspaces. A third method which combine the previous two is more efficient than
the other two enumerative encodings.Comment: 2009 Informaton Theory Workshop, Taormin
Schubert Varieties, Linear Codes and Enumerative Combinatorics
We consider linear error correcting codes associated to higher dimensional
projective varieties defined over a finite field. The problem of determining
the basic parameters of such codes often leads to some interesting and
difficult questions in combinatorics and algebraic geometry. This is
illustrated by codes associated to Schubert varieties in Grassmannians, called
Schubert codes, which have recently been studied. The basic parameters such as
the length, dimension and minimum distance of these codes are known only in
special cases. An upper bound for the minimum distance is known and it is
conjectured that this bound is achieved. We give explicit formulae for the
length and dimension of arbitrary Schubert codes and prove the minimum distance
conjecture in the affirmative for codes associated to Schubert divisors.Comment: 12 page
Large Constant Dimension Codes and Lexicodes
Constant dimension codes, with a prescribed minimum distance, have found
recently an application in network coding. All the codewords in such a code are
subspaces of \F_q^n with a given dimension. A computer search for large
constant dimension codes is usually inefficient since the search space domain
is extremely large. Even so, we found that some constant dimension lexicodes
are larger than other known codes. We show how to make the computer search more
efficient. In this context we present a formula for the computation of the
distance between two subspaces, not necessarily of the same dimension.Comment: submitted for ALCOMA1
Implementing Line-Hermitian Grassmann codes
In [I. Cardinali and L. Giuzzi. Line Hermitian Grassmann codes and their
parameters. Finite Fields Appl., 51: 407-432, 2018] we introduced line
Hermitian Grassmann codes and determined their parameters. The aim of this
paper is to present (in the spirit of [I. Cardinali and L. Giuzzi. Enumerative
coding for line polar Grassmannians with applications to codes. Finite Fields
Appl., 46:107-138, 2017]) an algorithm for the point enumerator of a line
Hermitian Grassmannian which can be usefully applied to get efficient encoders,
decoders and error correction algorithms for the aforementioned codes.Comment: 26 page
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