98,094 research outputs found
On the non-minimality of the largest weight codewords in the binary Reed-Muller codes
The study of minimal codewords in linear codes was motivated by Massey who described how minimal codewords of a linear code define access structures for secret sharing schemes. As a consequence of his article, Borissov, Manev, and Nikova initiated the study of minimal codewords in the binary Reed-Muller codes. They counted the number of non-minimal codewords of weight 2d in the binary Reed-Muller codes RM(r, in), and also gave results on the non-minimality of codewords of large weight in the binary Reed-Muller codes RM(r, in). The results of Borissov, Manev, and Nikova regarding the counting of the number of non-minimal codewords of small weight in RM(r,m) were improved by Schillewaert, Storme, and Thas who counted the number of non-minimal codewords of weight smaller than 3d in RM(r,m). This article now presents new results on the non-minimality of large weight codewords in RM(r, m)
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
On the number of minimal codewords in codes generated by the adjacency matrix of a graph
Minimal codewords have applications in decoding linear codes and in
cryptography. We study the number of minimal codewords in binary linear codes
that arise by appending a unit matrix to the adjacency matrix of a graph.Comment: 11 page
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
On Coset Leader Graphs of LDPC Codes
Our main technical result is that, in the coset leader graph of a linear
binary code of block length n, the metric balls spanned by constant-weight
vectors grow exponentially slower than those in .
Following the approach of Friedman and Tillich (2006), we use this fact to
improve on the first linear programming bound on the rate of LDPC codes, as the
function of their minimal distance. This improvement, combined with the
techniques of Ben-Haim and Lytsin (2006), improves the rate vs distance bounds
for LDPC codes in a significant sub-range of relative distances
Absolute type shaft encoding using LFSR
Maximal-length binary sequences have been known for a long time. They have many interesting properties, one of them is that when taken in blocks of n consecutive positions they form 2âż-1 different codes in a closed circular sequence. This property can be used for measuring absolute angular positions as the circle can be divided in as many parts as different codes can be retrieved. This paper describes how can a closed binary sequence with arbitrary length be effectively designed with the minimal possible block-length, using linear feedback shift registers (LFSR). Such sequences can be used for measuring a specified exact number of angular positions, using the minimal possible number of sensors that linear methods allow
Weight Spectrum of Quasi-Perfect Binary Codes with Distance 4
We consider the weight spectrum of a class of quasi-perfect binary linear
codes with code distance 4. For example, extended Hamming code and Panchenko
code are the known members of this class. Also, it is known that in many cases
Panchenko code has the minimal number of weight 4 codewords. We give exact
recursive formulas for the weight spectrum of quasi-perfect codes and their
dual codes. As an example of application of the weight spectrum we derive a
lower estimate for the conditional probability of correction of erasure
patterns of high weights (equal to or greater than code distance).Comment: 5 pages, 11 references, 2 tables; some explanations and detail are
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