52 research outputs found
Optimal Binary Locally Repairable Codes via Anticodes
This paper presents a construction for several families of optimal binary
locally repairable codes (LRCs) with small locality (2 and 3). This
construction is based on various anticodes. It provides binary LRCs which
attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal
with respect to the Griesmer bound
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
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
New Lower Bounds for Constant Dimension Codes
This paper provides new constructive lower bounds for constant dimension
codes, using different techniques such as Ferrers diagram rank metric codes and
pending blocks. Constructions for two families of parameters of constant
dimension codes are presented. The examples of codes obtained by these
constructions are the largest known constant dimension codes for the given
parameters
- β¦