8 research outputs found
The maximum number of minimal codewords in an code
Upper and lower bounds are derived for the quantity in the title, which is
tabulated for modest values of and An application to graphs with many
cycles is given.Comment: 6 pp. Submitte
On the minimum number of minimal codewords
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.Comment: 8 pages, 1 tabl
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
The maximum number of minimal codewords in long codes
AbstractUpper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provides lower bounds. In this paper, we compare these bounds with analogous bounds for the cycle code of graphs. This problem (in the graphic case) was considered in 1981 by Entringer and Slater who asked if a connected graph with p vertices and q edges can have only slightly more than 2q−p cycles. The bounds in this note answer this in the affirmative for all graphs except possibly some that have fewer than 2p+3log2(3p) edges. We also conclude that an Eulerian (even and connected) graph has at most 2q−p cycles unless the graph is a subdivision of a 4-regular graph that is the edge-disjoint union of two Hamiltonian cycles, in which case it may have as many as 2q−p+p cycles