MinT - Best Known (66−39, 66, s)-Nets in Base 32

Best Known (66−39, 66, s)-Nets in Base 32

(66−39, 66, 142)-Net over F₃₂ — Constructive and digital

Digital (27, 66, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 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, 46, 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]

(66−39, 66, 225)-Net over F₃₂ — Digital

Digital (27, 66, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 66, 225)-net over F₃₂, using
- net from sequence [i] based on digital (24, 224)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 24 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(66−39, 66, 259)-Net in Base 32 — Constructive

(27, 66, 259)-net in base 32, using

base change [i] based on (16, 55, 259)-net in base 64, using
- 1 times m-reduction [i] based on (16, 56, 259)-net in base 64, using
  - base change [i] based on digital (2, 42, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(66−39, 66, 321)-Net in Base 32

(27, 66, 321)-net in base 32, using

base change [i] based on (16, 55, 321)-net in base 64, using
- 1 times m-reduction [i] based on (16, 56, 321)-net in base 64, using
  - base change [i] based on digital (2, 42, 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]

(66−39, 66, 36052)-Net in Base 32 — Upper bound on s

There is no (27, 66, 36053)-net in base 32, because

1 times m-reduction [i] would yield (27, 65, 36053)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 68 369758 145162 306346 198413 780652 084634 628338 169476 645574 914647 565805 487417 985988 962070 279014 943772 > 32⁶⁵ [i]