Rank over Finite Fields and Codes

Abstract

Although the bilinear complexity of a bilinear map ϕ over a finite field may not be the minimum number of multiplications and divisions necessary for computing ϕ, the study of such maps gives some insight into the problem of computing a bilinear map over the ring of integers of a global field, such as the ring Z of integers: any bilinear computation defined over Z (that is, whose coefficients belong to Z) gives via reduction of constants modulo a prime p a bilinear computation over the finite field F _p . In this chapter we introduce a relationship observed by Brockett and Dobkin [80] between the rank of bilinear maps over a finite field and the theory of linear error-correcting codes. More precisely, we associate to any bilinear computation of length r of a bilinear map over a finite field a linear code of block length r; the dimension and minimum distance of this code depend only on the bilinear map and not on the specific computation. The question about lower bounds for r can then be stated as the question about the minimum block length of a linear code of given dimension and minimum distance. This question has been extensively studied by coding theorists; we use their results to obtain linear lower bounds for different problems, such as polynomial and matrix multiplication. In particular, following Bshouty [85, 86] we show that the rank of n x n-matrix multiplication over F₂ is 5/2n² — o(n²). In the last section of this chapter we discuss an interpolation algorithm on algebraic curves due to Chudnovsky and Chudnovsky [110]. Combined with a result on algebraic curves with many rational points over finite fields, this algorithm yields a linear upper bound for R(F _q ⁿ/F _q )for fixed q.