MinT - Best Known (56, 56+39, s)-Nets in Base 32

Best Known (56, 56+39, s)-Nets in Base 32

(56, 56+39, 300)-Net over F₃₂ — Constructive and digital

Digital (56, 95, 300)-net over F₃₂, using

2 times m-reduction [i] based on digital (56, 97, 300)-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 (9, 50, 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]

(56, 56+39, 515)-Net in Base 32 — Constructive

(56, 95, 515)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (12, 31, 257)-net in base 32, using
  - 1 times m-reduction [i] based on (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, 64, 258)-net in base 32, using
  - base change [i] based on digital (1, 40, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(56, 56+39, 2826)-Net over F₃₂ — Digital

Digital (56, 95, 2826)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+39, 7151295)-Net in Base 32 — Upper bound on s

There is no (56, 95, 7151296)-net in base 32, because

1 times m-reduction [i] would yield (56, 94, 7151296)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3048 589756 328276 603531 030883 712773 111756 431283 071834 304861 472371 003679 896087 306510 411412 857364 290150 851261 380877 901172 175909 823011 153520 835901 > 32⁹⁴ [i]