MinT - Best Known (37, 37+61, s)-Nets in Base 32

Best Known (37, 37+61, s)-Nets in Base 32

(37, 37+61, 131)-Net over F₃₂ — Constructive and digital

Digital (37, 98, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 30, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 68, 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]

(37, 37+61, 273)-Net over F₃₂ — Digital

Digital (37, 98, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 98, 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]

(37, 37+61, 288)-Net in Base 32 — Constructive

(37, 98, 288)-net in base 32, using

base change [i] based on digital (9, 70, 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]

(37, 37+61, 342)-Net in Base 32

(37, 98, 342)-net in base 32, using

4 times m-reduction [i] based on (37, 102, 342)-net in base 32, using
- base change [i] based on digital (20, 85, 342)-net over F₆₄, using
  - net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
      - a curve from the manYPoints database [i]

(37, 37+61, 28565)-Net in Base 32 — Upper bound on s

There is no (37, 98, 28566)-net in base 32, because

1 times m-reduction [i] would yield (37, 97, 28566)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 99 936867 236432 783488 507381 063215 808131 151558 765132 454705 920191 210522 809214 846455 174861 812284 992939 827839 139157 344215 447715 569048 778210 880468 435520 > 32⁹⁷ [i]