MinT - Best Known (78−45, 78, s)-Nets in Base 32

Best Known (78−45, 78, s)-Nets in Base 32

(78−45, 78, 162)-Net over F₃₂ — Constructive and digital

Digital (33, 78, 162)-net over F₃₂, using

1 times m-reduction [i] based on digital (33, 79, 162)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 26, 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, 53, 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]

(78−45, 78, 273)-Net over F₃₂ — Digital

Digital (33, 78, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 78, 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]

(78−45, 78, 288)-Net in Base 32 — Constructive

(33, 78, 288)-net in base 32, using

6 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
- base change [i] based on digital (9, 60, 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]

(78−45, 78, 342)-Net in Base 32

(33, 78, 342)-net in base 32, using

base change [i] based on digital (20, 65, 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]

(78−45, 78, 54128)-Net in Base 32 — Upper bound on s

There is no (33, 78, 54129)-net in base 32, because

1 times m-reduction [i] would yield (33, 77, 54129)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 78 818672 504996 930691 142864 660802 784652 250250 580142 419739 744683 182449 516678 050389 158327 723735 840894 102618 325989 544464 > 32⁷⁷ [i]