On forbidden submatrices

On linear forbidden submatrices

In this paper we study the extremal problem of finding how many 1 entries an n by n 0-1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0-1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0-1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0-1 matrix) li...

Avoiding Forbidden Submatrices by Row Deletions

We initiate a systematic study of the Row Deletion(B) problem on matrices: For a fixed “forbidden submatrix” B, the question is, given an input matrix A (both A and B have entries chosen from a finite-size alphabet), to remove a minimum number of rows such that A has no submatrix which is equivalent to a row or column permutation of B. An application of this question can be found, e.g., in the ...

Forbidden submatrices in 0-1 matrices

Forbidden Submatrices: Some New Bounds and Constructions

We explore an extremal hypergraph problem for which both the vertices and edges are ordered. Given a hypergraph F (not necessarily simple), we consider how many edges a simple hypergraph (no repeated edges) on m vertices can have while forbidding F as a subhypergraph where both hypergraphs have fixed vertex and edge orderings. A hypergraph of n edges on m vertices can be encoded as an m × n (0,...

Complex Orthogonal Designs with Forbidden 2 × 2 Submatrices

Complex orthogonal designs (CODs) are used to construct space-time block codes. COD Oz with parameter [p, n, k] is a p × n matrix, where nonzero entries are filled by ±zi or ±z ∗ i , i = 1, 2, . . . , k, such that O H z Oz = (|z1| 2 + |z2| 2 + . . . + |zk| )In×n. Define Oz an M-type COD if and only if Oz does not contain submatrix (