MinT - Best Known (63, 63+45, s)-Nets in Base 32

Best Known (63, 63+45, s)-Nets in Base 32

(63, 63+45, 316)-Net over F₃₂ — Constructive and digital

Digital (63, 108, 316)-net over F₃₂, using

2 times m-reduction [i] based on digital (63, 110, 316)-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 (7, 30, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (11, 58, 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]

(63, 63+45, 514)-Net in Base 32 — Constructive

(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)

(63, 63+45, 2776)-Net over F₃₂ — Digital

Digital (63, 108, 2776)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+45, 6109317)-Net in Base 32 — Upper bound on s

There is no (63, 108, 6109318)-net in base 32, because

1 times m-reduction [i] would yield (63, 107, 6109318)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 112473 004080 994071 133109 648568 224765 160549 607782 416085 255416 678094 820085 720192 923884 515689 450925 082575 441623 521973 369695 253175 255365 773515 102792 643057 548763 822016 > 32¹⁰⁷ [i]