MinT - Best Known (2, ∞, s)-Nets in Base 32

Best Known (2, ∞, s)-Nets in Base 32

(2, ∞, 44)-Net over F₃₂ — Constructive and digital

Digital (2, m, 44)-net over F₃₂ for arbitrarily large m, using

net from sequence [i] based on digital (2, 43)-sequence over F₃₂, using
- t-expansion [i] based on digital (1, 43)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
    - a function field by SÃ©mirat [i]

(2, ∞, 53)-Net over F₃₂ — Digital

Digital (2, m, 53)-net over F₃₂ for arbitrarily large m, using

net from sequence [i] based on digital (2, 52)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 2 and N(F) ≥ 53, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(2, ∞, 90)-Net in Base 32 — Upper bound on s

There is no (2, m, 91)-net in base 32 for arbitrarily large m, because

m-reduction [i] would yield (2, 87, 91)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32⁸⁷, 91, S₃₂, 85), but
  - the linear programming bound shows that MÂ â‰¥ 3218 248804 644806 290772 103657 191613 826913 486259 137011 802066 676331 137713 410455 220883 164990 453062 699224 855954 368127 744971 436177 620882 948096 / 34443 > 32⁸⁷ [i]