5,513 research outputs found
Binary and Ternary Quasi-perfect Codes with Small Dimensions
The aim of this work is a systematic investigation of the possible parameters
of quasi-perfect (QP) binary and ternary linear codes of small dimensions and
preparing a complete classification of all such codes. First we give a list of
infinite families of QP codes which includes all binary, ternary and quaternary
codes known to is. We continue further with a list of sporadic examples of
binary and ternary QP codes. Later we present the results of our investigation
where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions
up to 13 are classified.Comment: 4 page
Local Testing for Membership in Lattices
Motivated by the structural analogies between point lattices and linear error-correcting codes, and by the mature theory on locally testable codes, we initiate a systematic study of local testing for membership in lattices. Testing membership in lattices is also motivated in practice, by applications to integer programming, error detection in lattice-based communication, and cryptography. Apart from establishing the conceptual foundations of lattice testing, our results include the following: 1. We demonstrate upper and lower bounds on the query complexity of local testing for the well-known family of code formula lattices. Furthermore, we instantiate our results with code formula lattices constructed from Reed-Muller codes, and obtain nearly-tight bounds. 2. We show that in order to achieve low query complexity, it is sufficient to design one-sided non-adaptive canonical tests. This result is akin to, and based on an analogous result for error-correcting codes due to Ben-Sasson et al. (SIAM J. Computing 35(1) pp1-21)
Covering -Symbol Metric Codes and the Generalized Singleton Bound
Symbol-pair codes were proposed for the application in high density storage
systems, where it is not possible to read individual symbols. Yaakobi, Bruck
and Siegel proved that the minimum pair-distance of binary linear cyclic codes
satisfies and introduced -symbol metric
codes in 2016. In this paper covering codes in -symbol metrics are
considered. Some examples are given to show that the Delsarte bound and the
Norse bound for covering codes in the Hamming metric are not true for covering
codes in the pair metric. We give the redundancy bound on covering radius of
linear codes in the -symbol metric and give some optimal codes attaining
this bound. Then we prove that there is no perfect linear symbol-pair code with
the minimum pair distance and there is no perfect -symbol metric code if
. Moreover a lot of cyclic and algebraic-geometric codes
are proved non-perfect in the -symbol metric. The covering radius of the
Reed-Solomon code in the -symbol metric is determined. As an application the
generalized Singleton bound on the sizes of list-decodable -symbol metric
codes is also presented. Then an upper bound on lengths of general MDS
symbol-pair codes is proved.Comment: 21 page
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