MinT - Best Known (64, 64+45, s)-Nets in Base 32

Best Known (64, 64+45, s)-Nets in Base 32

(64, 64+45, 322)-Net over F₃₂ — Constructive and digital

Digital (64, 109, 322)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 22, 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 (9, 31, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]
3. digital (11, 56, 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]

(64, 64+45, 514)-Net in Base 32 — Constructive

(64, 109, 514)-net in base 32, using

32¹ times duplication [i] based on (63, 108, 514)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (14, 36, 257)-net in base 32, using
    - base change [i] based on (8, 30, 257)-net in base 64, using
      - 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
        base change [i] based on digital (0, 24, 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. (27, 72, 257)-net in base 32, using
    - base change [i] based on digital (0, 45, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(64, 64+45, 3002)-Net over F₃₂ — Digital

Digital (64, 109, 3002)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+45, 7151690)-Net in Base 32 — Upper bound on s

There is no (64, 109, 7151691)-net in base 32, because

1 times m-reduction [i] would yield (64, 108, 7151691)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 599133 598503 260226 994194 707736 381672 044437 846313 823393 905843 801044 090409 722869 718927 479915 769805 730256 317462 558747 355441 182830 641104 393240 684927 325044 938715 559288 > 32¹⁰⁸ [i]