On Bilinear Complexity of Multiplying 2 × 2-Matrix by 2 × m-Matrix over Finite Field

Abstract

The problem of the least number of multiplications required to compute the product of a 2 × 2-matrix X and a 2 × m-matrix Y over an arbitrary finite field is considered by assuming that the elements of the matrices are independent variables. No commutativity of elements of matrix X with elements of matrix Y is assumed (i.e., bilinear complexity is considered). Upper bound \(\frac{{7m}}{2}\) for this problem over an arbitrary field is known. For two-element field, this bound is exact. Lower bound (3 + \(\frac{3}{{{K^2} + 2}}\)) m is obtained for the least number of multiplications in this problem over an arbitrary finite field with K elements.