41 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)
On metric regularity of Reed-Muller codes
In this work we study metric properties of the well-known family of binary
Reed-Muller codes. Let be an arbitrary subset of the Boolean cube, and
be the metric complement of -- the set of all vectors of the
Boolean cube at the maximal possible distance from . If the metric
complement of coincides with , then the set is called a
{\it metrically regular set}. The problem of investigating metrically regular
sets appeared when studying {\it bent functions}, which have important
applications in cryptography and coding theory and are also one of the earliest
examples of a metrically regular set. In this work we describe metric
complements and establish the metric regularity of the codes
and for .
Additionally, the metric regularity of the codes and
is proved. Combined with previous results by Tokareva N.
(2012) concerning duality of affine and bent functions, this establishes the
metric regularity of most Reed-Muller codes with known covering radius. It is
conjectured that all Reed-Muller codes are metrically regular.Comment: 29 page
A Combinatorial Commutative Algebra Approach to Complete Decoding
Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, GeometrÃa y TopologÃ
Private Information Retrieval: Combinatorics of the Star-Product Scheme
In coded private information retrieval (PIR), a user wants to download a file from a distributed storage system without revealing the identity of the file. We consider the setting where certain subsets of servers collude to deduce the identity of the requested file. These subsets form an abstract simplicial complex called the collusion pattern. In this thesis, we study the combinatorics of the general star-product scheme for PIR under the assumption that the distributed storage system is encoded using a repetition code
Free Resolutions Associated to Representable Matroids
As a matroid is naturally a simplicial complex, one can study its combinatorial properties via the associated Stanley-Reisner ideal and its corresponding free resolution. Using results by Johnsen and Verdure, we prove that a matroid is the dual to a perfect matroid design if and only if its corresponding Stanley-Reisner ideal has a pure free resolution, and, motivated by applications to their generalized Hamming weights, characterize free resolutions corresponding to the vector matroids of the parity check matrices of Reed-Solomon codes and certain BCH codes. Furthermore, using an inductive mapping cone argument, we construct a cellular resolution for the matroid duals to finite projective geometries and discuss consequences for finite affine geometries. Finally, we provide algorithms for computing such cellular resolutions explicitly
On the Hamming distance of linear codes over a finite chain ring
Let R be a finite chain ring (e.g. a Galois ring), K its residue field and C a linear code
over R. We prove that d(C), the Hamming distance of C, is d((C : α)), where (C : α) is a
submodule quotient, α is a certain element of R and — denotes the canonical projection
to K. These two codes also have the same set of minimal codeword supports. We explicitly
construct a generator matrix/polynomial of (C : α) from the generator matrix/polynomials
of C. We show that in general d(C) ≤ d(C) with equality for free codes (i.e. for free R-
submodules of Rn) and in particular for Hensel lifts of cyclic codes over K. Most of the codes
over rings described in the literature fall into this class.
We characterise MDS codes over R and prove several analogues of properties of MDS codes
over finite fields. We compute the Hamming weight enumerator of a free MDS code over R
An Algorithmic Reduction Theory for Binary Codes: LLL and more
In this article, we propose an adaptation of the algorithmic reduction theory of lattices to binary codes. This includes the celebrated LLL algorithm (Lenstra, Lenstra, Lovasz, 1982), as well as adaptations of associated algorithms such as the Nearest Plane Algorithm of Babai (1986). Interestingly, the adaptation of LLL to binary codes can be interpreted as an algorithmic version of the bound of Griesmer (1960) on the minimal distance of a code.
Using these algorithms, we demonstrate ---both with a heuristic analysis and in practice--- a small polynomial speed-up over the Information-Set Decoding algorithm of Lee and Brickell (1988) for random binary codes. This appears to be the first such speed-up that is not based on a time-memory trade-off.
The above speed-up should be read as a very preliminary example of the potential of a reduction theory for codes, for example in cryptanalysis. In constructive cryptography, this algorithmic reduction theory could for example also be helpful for designing trapdoor functions from codes
An algorithmic reduction theory for binary codes: LLL and more
In this article, we propose an adaptation of the algorithmic reduction theory of lattices to binary codes. This includes the celebrated LLL algorithm (Lenstra, Lenstra, Lovasz, 1982), as well as adaptations of associated algorithms such as the Nearest Plane Algorithm of Babai (1986). Interestingly, the adaptation of LLL to binary codes can be interpreted as an algorithmic version of the bound of Griesmer (1960) on the minimal distance of a code. Using these algorithms, we demonstrate —both with a heuristic analysis and in practice— a small polynomial speed-up over the Information-Set Decoding algorithm of Lee and Brickell (1988) for random binary codes. This appears to be the first such speed-up that is not based on a time-memory trade-off. The above speed-up should be read as a very preliminary example of the potential of a reduction theory for codes, for example in cryptanalysis