MinT - Best Known (26, 26+41, s)-Nets in Base 32

Best Known (26, 26+41, s)-Nets in Base 32

(26, 26+41, 128)-Net over F₃₂ — Constructive and digital

Digital (26, 67, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 23, 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, 44, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(26, 26+41, 225)-Net over F₃₂ — Digital

Digital (26, 67, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 67, 225)-net over F₃₂, using
- net from sequence [i] based on digital (24, 224)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 24 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(26, 26+41, 257)-Net in Base 32 — Constructive

(26, 67, 257)-net in base 32, using

32¹ times duplication [i] based on (25, 66, 257)-net in base 32, using
- base change [i] based on (14, 55, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (14, 56, 257)-net in base 64, using
    - base change [i] based on digital (0, 42, 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]

(26, 26+41, 24817)-Net in Base 32 — Upper bound on s

There is no (26, 67, 24818)-net in base 32, because

1 times m-reduction [i] would yield (26, 66, 24818)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2187 733686 804162 347929 920984 365428 595430 283089 850596 212486 936165 441997 015313 486321 941561 172490 822552 > 32⁶⁶ [i]