MinT - Best Known (55−33, 55, s)-Nets in Base 32

Best Known (55−33, 55, s)-Nets in Base 32

(55−33, 55, 128)-Net over F₃₂ — Constructive and digital

Digital (22, 55, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 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 (3, 36, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(55−33, 55, 185)-Net over F₃₂ — Digital

Digital (22, 55, 185)-net over F₃₂, using

t-expansion [i] based on digital (21, 55, 185)-net over F₃₂, using
- net from sequence [i] based on digital (21, 184)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 21 and N(F) ≥ 185, using
    - a curve from the manYPoints database [i]

(55−33, 55, 258)-Net in Base 32 — Constructive

(22, 55, 258)-net in base 32, using

1 times m-reduction [i] based on (22, 56, 258)-net in base 32, using
- base change [i] based on digital (1, 35, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(55−33, 55, 289)-Net in Base 32

(22, 55, 289)-net in base 32, using

1 times m-reduction [i] based on (22, 56, 289)-net in base 32, using
- base change [i] based on digital (1, 35, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(55−33, 55, 26358)-Net in Base 32 — Upper bound on s

There is no (22, 55, 26359)-net in base 32, because

1 times m-reduction [i] would yield (22, 54, 26359)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1897 331072 206000 539188 349543 894005 737343 304764 562447 932419 556973 289135 024591 197295 > 32⁵⁴ [i]