77 research outputs found
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
Linear codes with few weights from non-weakly regular plateaued functions
Linear codes with few weights have significant applications in secret sharing
schemes, authentication codes, association schemes, and strongly regular
graphs. There are a number of methods to construct linear codes, one of which
is based on functions. Furthermore, two generic constructions of linear codes
from functions called the first and the second generic constructions, have
aroused the research interest of scholars. Recently, in \cite{Nian}, Li and
Mesnager proposed two open problems: Based on the first and the second generic
constructions, respectively, construct linear codes from non-weakly regular
plateaued functions and determine their weight distributions.
Motivated by these two open problems, in this paper, firstly, based on the
first generic construction, we construct some three-weight and at most
five-weight linear codes from non-weakly regular plateaued functions and
determine the weight distributions of the constructed codes. Next, based on the
second generic construction, we construct some three-weight and at most
five-weight linear codes from non-weakly regular plateaued functions belonging
to (defined in this paper) and determine the weight
distributions of the constructed codes. We also give the punctured codes of
these codes obtained based on the second generic construction and determine
their weight distributions. Meanwhile, we obtain some optimal and almost
optimal linear codes. Besides, by the Ashikhmin-Barg condition, we have that
the constructed codes are minimal for almost all cases and obtain some secret
sharing schemes with nice access structures based on their dual codes.Comment: 52 pages, 34 table
A geometric characterization of minimal codes and their asymptotic performance
In this paper, we give a geometric characterization of minimal linear codes.
In particular, we relate minimal linear codes to cutting blocking sets,
introduced in a recent paper by Bonini and Borello. Using this
characterization, we derive some bounds on the length and the distance of
minimal codes, according to their dimension and the underlying field size.
Furthermore, we show that the family of minimal codes is asymptotically good.
Finally, we provide some geometrical constructions of minimal codes.Comment: 22 page
Minimal -ary codes from non-covering permutations
In this article, we propose several generic methods for constructing minimal linear codes over the field . The first construction uses the method of direct sum of an arbitrary function and a bent function to induce minimal codes with parameters and minimum distance larger than . For the first time, we provide a general construction of linear codes from a subclass of non-weakly regular plateaued functions, which partially answers an open problem posed in [22]. The second construction deals with a bent function and a subspace of suitable derivatives of , i.e., functions of the form for some . We also provide a sound generalization of the recently introduced concept of non-covering permutations [45]. Some important structural properties of this class of permutations are derived in this context. The most remarkable observation is that the class of non-covering permutations contains the class of APN power permutations (characterized by having two-to-one derivatives). Finally, the last general construction combines the previous two methods (direct sum, non-covering permutations and subspaces of derivatives) together with a bent function in the Maiorana-McFarland class to construct minimal codes (even those violating the Ashikhmin-Barg bound) with a larger dimension. This last method proves to be quite flexible since it can lead to several non-equivalent codes, depending to a great extent on the choice of the underlying non-covering permutation
A Novel Application of Boolean Functions with High Algebraic Immunity in Minimal Codes
Boolean functions with high algebraic immunity are important cryptographic
primitives in some stream ciphers. In this paper, two methodologies for
constructing binary minimal codes from sets, Boolean functions and vectorial
Boolean functions with high algebraic immunity are proposed. More precisely, a
general construction of new minimal codes using minimal codes contained in
Reed-Muller codes and sets without nonzero low degree annihilators is
presented. The other construction allows us to yield minimal codes from certain
subcodes of Reed-Muller codes and vectorial Boolean functions with high
algebraic immunity. Via these general constructions, infinite families of
minimal binary linear codes of dimension and length less than or equal to
are obtained. In addition, a lower bound on the minimum distance of
the proposed minimal linear codes is established. Conjectures and open problems
are also presented. The results of this paper show that Boolean functions with
high algebraic immunity have nice applications in several fields such as
symmetric cryptography, coding theory and secret sharing schemes
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