It is proved that for an arbitrary polynomial f(x)Zpn[X] of degree d the Boolean complexity of calculation of one its root (if it exists) equals O(dM(n(p))) for a fixed prime p and growing n, where (p) = remvoelog(2)p, and M(n) is the Boolean complexity of multiplication of two binary n-bit numbers. Given the known decomposition of this number into prime factors n = m(1)...m(k), mi=pini, i = 1,..., k, with fixed k and primes p(i), i = 1,..., k, and growing n, the Boolean complexity of calculation of one of solutions to the comparison f(x) = 0 mod n equals O(dM((n))). In particular, the same estimate is obtained for calculation of one root of any given degree in the residue ring Z(m). As a corollary, it is proved that the Boolean complexity of calculation of integer roots of a polynomial f(x) is equal to O-d(M(n)), where f(x)=adxd+ad-1xd-1+...+a0,aiZ , |a(i)| < 2(n), i = 0,..., d.

It was proved that the complexity of square root computation in the Galois field GF(3 (s) ), s = 2 (k) r, is equal to O(M(2 (k) )M(r)k + M(r) log(2) r) + 2 (k) kr (1+o(1)), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3 (s) ) is equal to O(M(2 (k) )M(r)) and O(M(2 (k) )M(r)) + r (1+o(1)), respectively. If the basis in the field GF(3 (r) ) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2 (k) kr (1+o(1)) and r (1+o(1)) can be omitted. For M(n) one may take n log(2) n psi(n) where psi(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O (r) (M (s) log (2) s) = s (log (2) s)(2) psi(s).

Certain classes of binary constant weight codes can be represented geometrically using linear structures in Euclidean space. The geometric treatment is concerned mostly with codes with minimum distance 2(w − 1), that is, where any two codewords coincide in at most one entry; an algebraic generalization of parts of the theory also applies to some codes without this property. The presented theorems lead to several improvements of the tables of lower bounds on A(n, d, w) maintained by E. M. Rains and N. J. A. Sloane, and the ones recently published by D. H. Smith, L. A. Hughes and S. Perkins. Some of these new codes can be proven optimal.

In 2006, Smith et al. published a new table of constant weight codes, updating existing tables originally created by Brouwer et al. This paper improves upon these results by filling in 9 missing constant weight codes, all of which are optimal by the second Johnson bound. This completes the tables for A(n,16,9)A(n,16,9) and A(n,18,10)A(n,18,10) up to n=63n=63 and corrects some A(n,14,8)A(n,14,8).