13 research outputs found
Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures
The min-rank of a graph was introduced by Haemers (1978) to bound the Shannon
capacity of a graph. This parameter of a graph has recently gained much more
attention from the research community after the work of Bar-Yossef et al.
(2006). In their paper, it was shown that the min-rank of a graph G
characterizes the optimal scalar linear solution of an instance of the Index
Coding with Side Information (ICSI) problem described by the graph G. It was
shown by Peeters (1996) that computing the min-rank of a general graph is an
NP-hard problem. There are very few known families of graphs whose min-ranks
can be found in polynomial time. In this work, we introduce a new family of
graphs with efficiently computed min-ranks. Specifically, we establish a
polynomial time dynamic programming algorithm to compute the min-ranks of
graphs having simple tree structures. Intuitively, such graphs are obtained by
gluing together, in a tree-like structure, any set of graphs for which the
min-ranks can be determined in polynomial time. A polynomial time algorithm to
recognize such graphs is also proposed.Comment: Accepted by Algorithmica, 30 page
On the Security of Index Coding with Side Information
Security aspects of the Index Coding with Side Information (ICSI) problem are
investigated. Building on the results of Bar-Yossef et al. (2006), the
properties of linear index codes are further explored. The notion of weak
security, considered by Bhattad and Narayanan (2005) in the context of network
coding, is generalized to block security. It is shown that the linear index
code based on a matrix , whose column space code has length ,
minimum distance and dual distance , is -block secure
(and hence also weakly secure) if the adversary knows in advance
messages, and is completely insecure if the adversary knows in advance more
than messages. Strong security is examined under the conditions that
the adversary: (i) possesses messages in advance; (ii) eavesdrops at most
transmissions; (iii) corrupts at most transmissions. We prove
that for sufficiently large , an optimal linear index code which is strongly
secure against such an adversary has length . Here
is a generalization of the min-rank over of the side
information graph for the ICSI problem in its original formulation in the work
of Bar- Yossef et al.Comment: 14 page
Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes
An optimal constant-composition or constant-weight code of weight has
linear size if and only if its distance is at least . When , the determination of the exact size of such a constant-composition or
constant-weight code is trivial, but the case of has been solved
previously only for binary and ternary constant-composition and constant-weight
codes, and for some sporadic instances.
This paper provides a construction for quasicyclic optimal
constant-composition and constant-weight codes of weight and distance
based on a new generalization of difference triangle sets. As a result,
the sizes of optimal constant-composition codes and optimal constant-weight
codes of weight and distance are determined for all such codes of
sufficiently large lengths. This solves an open problem of Etzion.
The sizes of optimal constant-composition codes of weight and distance
are also determined for all , except in two cases.Comment: 12 page
Optimal index codes with near-extreme rates
Abstract—The min-rank of a digraph was shown by Bar-Yossef et al. (2006) to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information (ICSI) problem. In this work, the graphs and digraphs of near-extreme min-ranks are characterized. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. It is also shown that the decision problem of whether a digraph has min-rank two is NP-complete. By contrast, the same question for graphs can be answered in polynomial time. I
On secure Index Coding with Side Information
Abstract—Security aspects of the Index Coding with Side Information (ICSI) problem are investigated. Building on the results of Bar-Yossef et al. (2006), the properties of linear index codes are further explored. The notion of weak security, considered by Bhattad and Narayanan (2005) in the context of network coding, is generalized to block security. It is shown that the linear index code based on a matrix L, whose column space code C(L) has length n, minimum distance d and dual distance d ⊥ , is (d − 1 − t)-block secure (and hence also weakly secure) if the adversary knows in advance t ≤ d − 2 messages, and is completely insecure if the adversary knows in advance more than n − d ⊥ messages. Strong security is examined under the conditions that the adversary: (i) possesses t messages in advance; (ii) eavesdrops at most µ transmissions; (iii) corrupts at most δ transmissions. We prove that for sufficiently large q, an optimal linear index code, which is strongly secure against such an adversary, has length κq +µ+2δ. Here κq is a generalization of the min-rank over Fq of the side information graph for the ICSI problem in its original formulation in the work of Bar-Yossef et al
Index coding and error correction
Abstract—A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols δ, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the α-bound and the κ-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Finally, the decoding of linear ECIC’s is discussed. The syndrome decoding is shown to output the exact message if the weight of the error vector is less or equal to the error-correcting capability of the corresponding ECIC