MinT - Best Known (42, 42+57, s)-Nets in Base 32

Best Known (42, 42+57, s)-Nets in Base 32

(42, 42+57, 196)-Net over F₃₂ — Constructive and digital

Digital (42, 99, 196)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 35, 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]
2. digital (7, 64, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(42, 42+57, 288)-Net in Base 32 — Constructive

(42, 99, 288)-net in base 32, using

t-expansion [i] based on (40, 99, 288)-net in base 32, using
- 9 times m-reduction [i] based on (40, 108, 288)-net in base 32, using
  - base change [i] based on (22, 90, 288)-net in base 64, using
    - 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 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]

(42, 42+57, 310)-Net over F₃₂ — Digital

Digital (42, 99, 310)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+57, 408)-Net in Base 32

(42, 99, 408)-net in base 32, using

3 times m-reduction [i] based on (42, 102, 408)-net in base 32, using
- base change [i] based on digital (25, 85, 408)-net over F₆₄, using
  - net from sequence [i] based on digital (25, 407)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 25 and N(F) ≥ 408, using
      - a curve from the manYPoints database [i]

(42, 42+57, 67542)-Net in Base 32 — Upper bound on s

There is no (42, 99, 67543)-net in base 32, because

1 times m-reduction [i] would yield (42, 98, 67543)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3197 635985 551849 532090 651071 030969 189367 695605 492690 523701 251787 627217 643635 970208 103848 919361 289997 034552 939746 426722 049609 989533 576675 192604 441464 > 32⁹⁸ [i]