MinT - Best Known (45−25, 45, s)-Nets in Base 32

Best Known (45−25, 45, s)-Nets in Base 32

(45−25, 45, 142)-Net over F₃₂ — Constructive and digital

Digital (20, 45, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 44)-net over F₃₂, using
  - net from sequence [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. digital (7, 32, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(45−25, 45, 221)-Net over F₃₂ — Digital

Digital (20, 45, 221)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(45−25, 45, 259)-Net in Base 32 — Constructive

(20, 45, 259)-net in base 32, using

3 times m-reduction [i] based on (20, 48, 259)-net in base 32, using
- base change [i] based on digital (2, 30, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(45−25, 45, 321)-Net in Base 32

(20, 45, 321)-net in base 32, using

3 times m-reduction [i] based on (20, 48, 321)-net in base 32, using
- base change [i] based on digital (2, 30, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(45−25, 45, 56342)-Net in Base 32 — Upper bound on s

There is no (20, 45, 56343)-net in base 32, because

1 times m-reduction [i] would yield (20, 44, 56343)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 685106 932102 171877 386057 175133 552871 434889 161497 088602 867157 314422 > 32⁴⁴ [i]