On Hadamard embeddability

AbstractGiven any k vectors of dimension nk which are mutually orthogonal, it is well known that this matrix can be completed to an n×n orthogonal matrix. Hadamard matrices form a subclass of orthogonal matrices. By contrast it is shown that it is possible to construct Hadamard submatrices with 2t+2 rows that cannot be completed to a Hadamard matrix of order 4t for infinitely many values of t. Some familiarity with Hasse–Minkowski invariants is assumed. A large number of unsolved problems in this area are pointed out