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.

中文翻译：

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有限域上2×2-矩阵乘2×m-矩阵的双线性复杂度

通过假定矩阵的元素是自变量，来考虑在任意有限域上计算2×2矩阵X和2× m矩阵Y的乘积所需的最小乘法数问题。不假设矩阵X的元素与矩阵Y的元素具有可交换性（即，考虑了双线性复杂度）。已知此问题在任意字段上的上限\（\ frac {{7m}} {2} \）。对于二元素字段，此界限是精确的。下界（3 + \（\ frac {3} {{{K ^ 2} + 2}} \））m是在任意有限域上对该问题的乘法次数最少的情况下获得的K个元素。