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

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

(27, 27+32, 162)-Net over F₃₂ — Constructive and digital

Digital (27, 59, 162)-net over F₃₂, using

2 times m-reduction [i] based on digital (27, 61, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 20, 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, 41, 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]

(27, 27+32, 288)-Net in Base 32 — Constructive

(27, 59, 288)-net in base 32, using

4 times m-reduction [i] based on (27, 63, 288)-net in base 32, using
- base change [i] based on digital (9, 45, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(27, 27+32, 303)-Net over F₃₂ — Digital

Digital (27, 59, 303)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+32, 321)-Net in Base 32

(27, 59, 321)-net in base 32, using

7 times m-reduction [i] based on (27, 66, 321)-net in base 32, using
- base change [i] based on (16, 55, 321)-net in base 64, using
  - 1 times m-reduction [i] based on (16, 56, 321)-net in base 64, using
    - base change [i] based on digital (2, 42, 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]

(27, 27+32, 77870)-Net in Base 32 — Upper bound on s

There is no (27, 59, 77871)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 63664 127505 820380 477370 849231 540616 313091 559491 094299 460829 563305 106300 862327 425084 024042 > 32⁵⁹ [i]