Related papers: On the number of minimal codewords in codes genera…
Minimal codewords have applications in decoding linear codes and in cryptography. We study the maximum number of minimal codewords in binary linear codes of a given length and dimension. Improved lower and upper bounds on the maximum number…
We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.
In this paper, we consider minimal linear codes in a general construction of linear codes from q-ary functions. First, we give the sufficient and necessary condition for codewords to be minimal. Second, as an application, we present four…
In order to understand the performance of a code under maximum-likelihood (ML) decoding, it is crucial to know the minimal codewords. In the context of linear programming (LP) decoding, it turns out to be necessary to know the minimal…
Upper and lower bounds are derived for the quantity in the title, which is tabulated for modest values of $n$ and $k.$ An application to graphs with many cycles is given.
Binary linear codes are constructed from graphs, in particular, by the generator matrix $[I_n|A]$ where $A$ is the adjacency matrix of a graph on $n$ vertices. A combinatorial interpretation of the minimum distance of such codes is given.…
In order to understand the performance of a code under maximum-likelihood (ML) decoding, one studies the codewords, in particular the minimal codewords, and their Hamming weights. In the context of linear programming (LP) decoding, one's…
Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provide lower bounds. In this paper we compare these bounds with analogous bounds for the cycle code of graphs. This…
There are several methods for constructing secret sharing schemes, one of which is based on coding theory. Theoretically, every linear code can be used to construct secret sharing schemes. However, in general, determining the access…
Partial spread is important in finite geometry and can be used to construct linear codes. From the results in (Designs, Codes and Cryptography 90:1-15, 2022) by Xia Li, Qin Yue and Deng Tang, we know that if the number of the elements in a…
In this paper we study spread codes: a family of constant-dimension codes for random linear network coding. In other words, the codewords are full-rank matrices of size (k x n) with entries in a finite field F_q. Spread codes are a family…
In this paper we show the usability of the Gray code with constant weight words for computing linear combinations of codewords. This can lead to a big improvement of the computation time for finding the minimum distance of a code. We have…
Linear-programming pseudocodewords play a pivotal role in our understanding of the linear-programming decoding algorithms. These pseudocodewords are known to be equivalent to the graph-cover pseudocodewords. The latter pseudocodewords, when…
A minimal code is a linear code where the only instance that a codeword has its support contained in the support of another codeword is when the codewords are scalar multiples of each other. Ashikhmin and Barg gave a sufficient condition…
We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues…
In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gr\"obner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for…
Let $k\leq n$ be two positive integers and $q$ a prime power. The basic question in minimal linear codes is to determine if there exists an $[n,k]_q$ minimal linear code. The first objective of this paper is to present a new sufficient and…
Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$…
Binary codes are constructed from incidence matrices of hypergraphs. A combinatroial description is given for the minimum distances of such codes via a combinatorial tool called ``eonv". This combinatorial approach provides a faster…
This papers presents a detailed analysis of pseudocodewords of Tanner graphs. Pseudocodewords arising on the iterative decoder's computation tree are distinguished from pseudocodewords arising on finite degree lifts. Lower bounds on the…