MinT - Best Known (68−39, 68, s)-Nets in Base 32

Best Known (68−39, 68, s)-Nets in Base 32

(68−39, 68, 162)-Net over F₃₂ — Constructive and digital

Digital (29, 68, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 22, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, 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]

(68−39, 68, 257)-Net over F₃₂ — Digital

Digital (29, 68, 257)-net over F₃₂, using

t-expansion [i] based on digital (28, 68, 257)-net over F₃₂, using
- net from sequence [i] based on digital (28, 256)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 28 and N(F) ≥ 257, using
    - a curve from the manYPoints database [i]

(68−39, 68, 288)-Net in Base 32 — Constructive

(29, 68, 288)-net in base 32, using

2 times m-reduction [i] based on (29, 70, 288)-net in base 32, using
- base change [i] based on digital (9, 50, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(68−39, 68, 321)-Net in Base 32

(29, 68, 321)-net in base 32, using

4 times m-reduction [i] based on (29, 72, 321)-net in base 32, using
- base change [i] based on digital (2, 45, 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]

(68−39, 68, 51928)-Net in Base 32 — Upper bound on s

There is no (29, 68, 51929)-net in base 32, because

1 times m-reduction [i] would yield (29, 67, 51929)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 69999 363482 994475 839101 094180 486055 713883 278260 395524 062480 955602 282116 326736 975191 441765 717218 410776 > 32⁶⁷ [i]