Generalizations and applications of the nowhere zero linear mappings in network coding (Q616419)

A vector is nowhere zero if it contains no zero component. Suppose that \(A\) is an \(n \times k\) matrix \(A\) over a field \(\mathbf F\) on \(q \geq n+2\) elements. The main result of the paper establishes that whenever \(A\) contains no zero row, there is a nowhere zero vector \(x\) so that \(Ax\) is nowhere zero. An algorithm is developed to find such a nowhere zero vector \(x\) given the matrix \(A\). This in turn leads to an efficient algorithm for the algebraic construction of acyclic network codes.