MinT - Best Known (55, 55+39, s)-Nets in Base 32

Best Known (55, 55+39, s)-Nets in Base 32

(55, 55+39, 300)-Net over F₃₂ — Constructive and digital

Digital (55, 94, 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, 26, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (9, 48, 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]

(55, 55+39, 514)-Net in Base 32 — Constructive

(55, 94, 514)-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. (24, 63, 257)-net in base 32, using
  - 1 times m-reduction [i] based on (24, 64, 257)-net in base 32, using
    - base change [i] based on digital (0, 40, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(55, 55+39, 2582)-Net over F₃₂ — Digital

Digital (55, 94, 2582)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+39, 5958900)-Net in Base 32 — Upper bound on s

There is no (55, 94, 5958901)-net in base 32, because

1 times m-reduction [i] would yield (55, 93, 5958901)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 95 268232 294919 211446 705533 645105 919291 646867 380453 871453 783914 222238 040064 039158 273680 985418 390630 788285 011582 555780 293323 093017 135634 766948 > 32⁹³ [i]