MinT - Best Known (31, 72, s)-Nets in Base 32

Best Known (31, 72, s)-Nets in Base 32

(31, 72, 162)-Net over F₃₂ — Constructive and digital

Digital (31, 72, 162)-net over F₃₂, using

1 times m-reduction [i] based on digital (31, 73, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 24, 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, 49, 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]

(31, 72, 273)-Net over F₃₂ — Digital

Digital (31, 72, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 72, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(31, 72, 288)-Net in Base 32 — Constructive

(31, 72, 288)-net in base 32, using

5 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
- base change [i] based on digital (9, 55, 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]

(31, 72, 321)-Net in Base 32

(31, 72, 321)-net in base 32, using

5 times m-reduction [i] based on (31, 77, 321)-net in base 32, using
- base change [i] based on (9, 55, 321)-net in base 128, using
  - 1 times m-reduction [i] based on (9, 56, 321)-net in base 128, using
    - base change [i] based on digital (2, 49, 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]

(31, 72, 59040)-Net in Base 32 — Upper bound on s

There is no (31, 72, 59041)-net in base 32, because

1 times m-reduction [i] would yield (31, 71, 59041)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 73396 556713 448196 157389 947938 718471 746402 961765 923504 524039 374723 391334 132902 354645 221553 726540 035777 269672 > 32⁷¹ [i]