51,376 research outputs found
Artin-Schreier families and 2-D cycle codes
We start with the study of certain Artin-Schreier families. Using coding theory techniques, we determine a necessary and sufficient condition for such families to have a nontrivial curve with the maximum possible number of rational points over the finite field in consideration. This result produces several nice corollaries, including the existence of certain maximal curves; i.e., curves meeting the Hasse-Weil bound.We then present a way to represent two-dimensional (2-D) cyclic codes as trace codes starting from a basic zero set of its dual code. This representation enables us to relate the weight of a codeword to the number of rational points on certain Artin-Schreier curves via the additive form of Hilbert’s Theorem 90. We use our results on Artin-Schreier families to give a minimum distance bound for a large class of 2-D cyclic codes. Then, we look at some specific classes of 2-D cyclic codes that are not covered by our general result. In one case, we obtain the complete weight enumerator and show that these types of codes have two nonzero weights. In the other cases, we again give minimum distance bounds. We present examples, in some of which our estimates are fairly effcient
General Scheme for Perfect Quantum Network Coding with Free Classical Communication
This paper considers the problem of efficiently transmitting quantum states
through a network. It has been known for some time that without additional
assumptions it is impossible to achieve this task perfectly in general --
indeed, it is impossible even for the simple butterfly network. As additional
resource we allow free classical communication between any pair of network
nodes. It is shown that perfect quantum network coding is achievable in this
model whenever classical network coding is possible over the same network when
replacing all quantum capacities by classical capacities. More precisely, it is
proved that perfect quantum network coding using free classical communication
is possible over a network with source-target pairs if there exists a
classical linear (or even vector linear) coding scheme over a finite ring. Our
proof is constructive in that we give explicit quantum coding operations for
each network node. This paper also gives an upper bound on the number of
classical communication required in terms of , the maximal fan-in of any
network node, and the size of the network.Comment: 12 pages, 2 figures, generalizes some of the results in
arXiv:0902.1299 to the k-pair problem and codes over rings. Appeared in the
Proceedings of the 36th International Colloquium on Automata, Languages and
Programming (ICALP'09), LNCS 5555, pp. 622-633, 200
On the group of a rational maximal bifix code
We give necessary and sufficient conditions for the group of a rational
maximal bifix code to be isomorphic with the -group of , when
is recurrent and is rational. The case where is uniformly
recurrent, which is known to imply the finiteness of , receives
special attention.
The proofs are done by exploring the connections with the structure of the
free profinite monoid over the alphabet of
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
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