MinT - Best Known (77−43, 77, s)-Nets in Base 32

Best Known (77−43, 77, s)-Nets in Base 32

(77−43, 77, 174)-Net over F₃₂ — Constructive and digital

Digital (34, 77, 174)-net over F₃₂, using

1 times m-reduction [i] based on digital (34, 78, 174)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 27, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]
  2. digital (7, 51, 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]

(77−43, 77, 288)-Net in Base 32 — Constructive

(34, 77, 288)-net in base 32, using

t-expansion [i] based on (33, 77, 288)-net in base 32, using
- 7 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]

(77−43, 77, 305)-Net over F₃₂ — Digital

Digital (34, 77, 305)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−43, 77, 342)-Net in Base 32

(34, 77, 342)-net in base 32, using

7 times m-reduction [i] based on (34, 84, 342)-net in base 32, using
- base change [i] based on digital (20, 70, 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]

(77−43, 77, 78393)-Net in Base 32 — Upper bound on s

There is no (34, 77, 78394)-net in base 32, because

1 times m-reduction [i] would yield (34, 76, 78394)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 462822 090775 007645 810630 923820 324084 854578 894745 768491 726063 889810 418165 790454 814580 377825 669540 327247 050204 352610 > 32⁷⁶ [i]