3,202 research outputs found
New binary and ternary LCD codes
LCD codes are linear codes with important cryptographic applications.
Recently, a method has been presented to transform any linear code into an LCD
code with the same parameters when it is supported on a finite field with
cardinality larger than 3. Hence, the study of LCD codes is mainly open for
binary and ternary fields. Subfield-subcodes of -affine variety codes are a
generalization of BCH codes which have been successfully used for constructing
good quantum codes. We describe binary and ternary LCD codes constructed as
subfield-subcodes of -affine variety codes and provide some new and good LCD
codes coming from this construction
Linear Codes from Some 2-Designs
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -design as a generator
matrix over \gf(q) of the code. This approach has been extensively
investigated in the literature. In this paper, a different method of
constructing linear codes using specific classes of -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -designs associated to semibent functions.
Two families of the codes obtained in this paper are optimal. The linear codes
presented in this paper have applications in secret sharing and authentication
schemes, in addition to their applications in consumer electronics,
communication and data storage systems. A coding-theory approach to the
characterisation of highly nonlinear Boolean functions is presented
High-rate self-synchronizing codes
Self-synchronization under the presence of additive noise can be achieved by
allocating a certain number of bits of each codeword as markers for
synchronization. Difference systems of sets are combinatorial designs which
specify the positions of synchronization markers in codewords in such a way
that the resulting error-tolerant self-synchronizing codes may be realized as
cosets of linear codes. Ideally, difference systems of sets should sacrifice as
few bits as possible for a given code length, alphabet size, and
error-tolerance capability. However, it seems difficult to attain optimality
with respect to known bounds when the noise level is relatively low. In fact,
the majority of known optimal difference systems of sets are for exceptionally
noisy channels, requiring a substantial amount of bits for synchronization. To
address this problem, we present constructions for difference systems of sets
that allow for higher information rates while sacrificing optimality to only a
small extent. Our constructions utilize optimal difference systems of sets as
ingredients and, when applied carefully, generate asymptotically optimal ones
with higher information rates. We also give direct constructions for optimal
difference systems of sets with high information rates and error-tolerance that
generate binary and ternary self-synchronizing codes.Comment: 9 pages, no figure, 2 tables. Final accepted version for publication
in the IEEE Transactions on Information Theory. Material presented in part at
the International Symposium on Information Theory and its Applications,
Honolulu, HI USA, October 201
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