MinT - Best Known (58−35, 58, s)-Nets in Base 32

Best Known (58−35, 58, s)-Nets in Base 32

(58−35, 58, 128)-Net over F₃₂ — Constructive and digital

Digital (23, 58, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 20, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 38, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(58−35, 58, 185)-Net over F₃₂ — Digital

Digital (23, 58, 185)-net over F₃₂, using

t-expansion [i] based on digital (21, 58, 185)-net over F₃₂, using
- net from sequence [i] based on digital (21, 184)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 21 and N(F) ≥ 185, using
    - a curve from the manYPoints database [i]

(58−35, 58, 257)-Net in Base 32 — Constructive

(23, 58, 257)-net in base 32, using

2 times m-reduction [i] based on (23, 60, 257)-net in base 32, using
- base change [i] based on (13, 50, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (13, 52, 257)-net in base 64, using
    - base change [i] based on digital (0, 39, 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]

(58−35, 58, 25770)-Net in Base 32 — Upper bound on s

There is no (23, 58, 25771)-net in base 32, because

1 times m-reduction [i] would yield (23, 57, 25771)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 62 195688 732463 526530 720677 599760 336790 833288 859340 339087 816255 703304 105384 381675 981884 > 32⁵⁷ [i]