MinT - Best Known (50−27, 50, s)-Nets in Base 32

Best Known (50−27, 50, s)-Nets in Base 32

(50−27, 50, 162)-Net over F₃₂ — Constructive and digital

Digital (23, 50, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 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, 34, 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]

(50−27, 50, 260)-Net in Base 32 — Constructive

(23, 50, 260)-net in base 32, using

32² times duplication [i] based on (21, 48, 260)-net in base 32, using
- base change [i] based on digital (3, 30, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(50−27, 50, 280)-Net over F₃₂ — Digital

Digital (23, 50, 280)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(50−27, 50, 321)-Net in Base 32

(23, 50, 321)-net in base 32, using

6 times m-reduction [i] based on (23, 56, 321)-net in base 32, using
- base change [i] based on digital (2, 35, 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]

(50−27, 50, 86152)-Net in Base 32 — Upper bound on s

There is no (23, 50, 86153)-net in base 32, because

1 times m-reduction [i] would yield (23, 49, 86153)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 56 540236 625416 081934 264055 873481 381603 075550 846078 902797 242794 494267 487872 > 32⁴⁹ [i]