MinT - Best Known (51−29, 51, s)-Nets in Base 32

Best Known (51−29, 51, s)-Nets in Base 32

(51−29, 51, 142)-Net over F₃₂ — Constructive and digital

Digital (22, 51, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 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, 36, 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]

(51−29, 51, 205)-Net over F₃₂ — Digital

Digital (22, 51, 205)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−29, 51, 258)-Net in Base 32 — Constructive

(22, 51, 258)-net in base 32, using

5 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]

(51−29, 51, 289)-Net in Base 32

(22, 51, 289)-net in base 32, using

5 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]

(51−29, 51, 46298)-Net in Base 32 — Upper bound on s

There is no (22, 51, 46299)-net in base 32, because

1 times m-reduction [i] would yield (22, 50, 46299)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1809 550554 161132 107517 293532 332550 104935 262133 962503 197962 495127 663607 922292 > 32⁵⁰ [i]