131,565 research outputs found
Minimal linear codes from characteristic functions
Minimal linear codes have interesting applications in secret sharing schemes
and secure two-party computation. This paper uses characteristic functions of
some subsets of to construct minimal linear codes. By properties
of characteristic functions, we can obtain more minimal binary linear codes
from known minimal binary linear codes, which generalizes results of Ding et
al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By
characteristic functions corresponding to some subspaces of , we
obtain many minimal linear codes, which generalizes results of [IEEE Trans.
Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf.
Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic
functions to present a characterization of minimal linear codes from the
defining set method and present a class of minimal linear codes
A new class of three-weight linear codes from weakly regular plateaued functions
Linear codes with few weights have many applications in secret sharing
schemes, authentication codes, communication and strongly regular graphs. In
this paper, we consider linear codes with three weights in arbitrary
characteristic. To do this, we generalize the recent contribution of Mesnager
given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present
a new class of binary linear codes with three weights from plateaued Boolean
functions and their weight distributions. We next introduce the notion of
(weakly) regular plateaued functions in odd characteristic and give
concrete examples of these functions. Moreover, we construct a new class of
three-weight linear -ary codes from weakly regular plateaued functions and
determine their weight distributions. We finally analyse the constructed linear
codes for secret sharing schemes.Comment: The Extended Abstract of this work was submitted to WCC-2017 (the
Tenth International Workshop on Coding and Cryptography
Binary Cyclic Codes from Explicit Polynomials over \gf(2^m)
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, monomials and
trinomials over finite fields with even characteristic are employed to
construct a number of families of binary cyclic codes. Lower bounds on the
minimum weight of some families of the cyclic codes are developed. The minimum
weights of other families of the codes constructed in this paper are
determined. The dimensions of the codes are flexible. Some of the codes
presented in this paper are optimal or almost optimal in the sense that they
meet some bounds on linear codes. Open problems regarding binary cyclic codes
from monomials and trinomials are also presented.Comment: arXiv admin note: substantial text overlap with arXiv:1206.4687,
arXiv:1206.437
On decomposability of 4-ary distance 2 MDS codes, double-codes, and n-quasigroups of order 4
A subset of is called a -fold MDS code if every
line in each of base directions contains exactly elements of . The
adjacency graph of a -fold MDS code is not connected if and only if the
characteristic function of the code is the repetition-free sum of the
characteristic functions of -fold MDS codes of smaller lengths.
In the case , the theory has the following application. The union of two
disjoint MDS codes in is a double-MDS-code. If
the adjacency graph of the double-MDS-code is not connected, then the
double-code can be decomposed into double-MDS-codes of smaller lengths. If the
graph has more than two connected components, then the MDS codes are also
decomposable. The result has an interpretation as a test for reducibility of
-quasigroups of order 4. Keywords: MDS codes, n-quasigroups,
decomposability, reducibility, frequency hypercubes, latin hypercubesComment: 19 pages. V2: revised, general case q=2t is added. Submitted to
Discr. Mat
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