MinT - Best Known (84−35, 84, s)-Nets in Base 32

Best Known (84−35, 84, s)-Nets in Base 32

(84−35, 84, 294)-Net over F₃₂ — Constructive and digital

Digital (49, 84, 294)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 18, 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, 24, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (7, 42, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(84−35, 84, 514)-Net in Base 32 — Constructive

(49, 84, 514)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (11, 28, 257)-net in base 32, using
  - base change [i] based on (3, 20, 257)-net in base 128, using
    - 4 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
      - base change [i] based on digital (0, 21, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. (21, 56, 257)-net in base 32, using
  - base change [i] based on digital (0, 35, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(84−35, 84, 2301)-Net over F₃₂ — Digital

Digital (49, 84, 2301)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−35, 84, 5167211)-Net in Base 32 — Upper bound on s

There is no (49, 84, 5167212)-net in base 32, because

1 times m-reduction [i] would yield (49, 83, 5167212)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 84615 369648 913621 161984 723201 707048 246628 919556 193791 628786 829022 391357 374291 349075 698643 013124 933173 553942 379851 099945 844814 > 32⁸³ [i]