MinT - Best Known (25, 25+37, s)-Nets in Base 32

Best Known (25, 25+37, s)-Nets in Base 32

(25, 25+37, 131)-Net over F₃₂ — Constructive and digital

Digital (25, 62, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 44, 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]

(25, 25+37, 225)-Net over F₃₂ — Digital

Digital (25, 62, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 62, 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]

(25, 25+37, 258)-Net in Base 32 — Constructive

(25, 62, 258)-net in base 32, using

2 times m-reduction [i] based on (25, 64, 258)-net in base 32, using
- base change [i] based on digital (1, 40, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(25, 25+37, 289)-Net in Base 32

(25, 62, 289)-net in base 32, using

2 times m-reduction [i] based on (25, 64, 289)-net in base 32, using
- base change [i] based on digital (1, 40, 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]

(25, 25+37, 30720)-Net in Base 32 — Upper bound on s

There is no (25, 62, 30721)-net in base 32, because

1 times m-reduction [i] would yield (25, 61, 30721)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 65 210257 451964 080410 882069 038367 256442 960627 524617 541927 991433 273430 052497 011292 477824 672768 > 32⁶¹ [i]