MinT - Best Known (53−30, 53, s)-Nets in Base 32

Best Known (53−30, 53, s)-Nets in Base 32

(53−30, 53, 142)-Net over F₃₂ — Constructive and digital

Digital (23, 53, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 16, 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, 37, 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]

(53−30, 53, 216)-Net over F₃₂ — Digital

Digital (23, 53, 216)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(53−30, 53, 259)-Net in Base 32 — Constructive

(23, 53, 259)-net in base 32, using

3 times m-reduction [i] based on (23, 56, 259)-net in base 32, using
- base change [i] based on digital (2, 35, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(53−30, 53, 321)-Net in Base 32

(23, 53, 321)-net in base 32, using

3 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]

(53−30, 53, 43104)-Net in Base 32 — Upper bound on s

There is no (23, 53, 43105)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 59 288631 341527 477318 440266 607074 913408 217055 200212 585294 736853 197157 110078 760928 > 32⁵³ [i]