MinT - Best Known (99−41, 99, s)-Nets in Base 32

Best Known (99−41, 99, s)-Nets in Base 32

(99−41, 99, 316)-Net over F₃₂ — Constructive and digital

Digital (58, 99, 316)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 20, 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, 27, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (11, 52, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(99−41, 99, 514)-Net in Base 32 — Constructive

(58, 99, 514)-net in base 32, using

32¹ times duplication [i] based on (57, 98, 514)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (12, 32, 257)-net in base 32, using
    - base change [i] based on digital (0, 20, 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. (25, 66, 257)-net in base 32, using
    - base change [i] based on (14, 55, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (14, 56, 257)-net in base 64, using
        base change [i] based on digital (0, 42, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(99−41, 99, 2722)-Net over F₃₂ — Digital

Digital (58, 99, 2722)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−41, 99, 6355924)-Net in Base 32 — Upper bound on s

There is no (58, 99, 6355925)-net in base 32, because

1 times m-reduction [i] would yield (58, 98, 6355925)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3196 674389 622823 026873 665573 894576 725021 826679 267672 684734 918306 246071 562545 881057 763900 133616 764846 155289 467244 478320 858451 897641 957342 952205 149541 > 32⁹⁸ [i]