MinT - Best Known (29, 29+45, s)-Nets in Base 32

Best Known (29, 29+45, s)-Nets in Base 32

(29, 29+45, 131)-Net over F₃₂ — Constructive and digital

Digital (29, 74, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 52, 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]

(29, 29+45, 257)-Net in Base 32 — Constructive

(29, 74, 257)-net in base 32, using

3 times m-reduction [i] based on (29, 77, 257)-net in base 32, using
- base change [i] based on (7, 55, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (7, 56, 257)-net in base 128, using
    - base change [i] based on digital (0, 49, 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]

(29, 29+45, 257)-Net over F₃₂ — Digital

Digital (29, 74, 257)-net over F₃₂, using

t-expansion [i] based on digital (28, 74, 257)-net over F₃₂, using
- net from sequence [i] based on digital (28, 256)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 28 and N(F) ≥ 257, using
    - a curve from the manYPoints database [i]

(29, 29+45, 267)-Net in Base 32

(29, 74, 267)-net in base 32, using

4 times m-reduction [i] based on (29, 78, 267)-net in base 32, using
- base change [i] based on digital (16, 65, 267)-net over F₆₄, using
  - net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
      - a curve from the manYPoints database [i]

(29, 29+45, 28819)-Net in Base 32 — Upper bound on s

There is no (29, 74, 28820)-net in base 32, because

1 times m-reduction [i] would yield (29, 73, 28820)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 75 192300 552209 750332 554694 524843 010462 830333 952256 173496 654519 278635 895768 069483 294195 193051 792418 074521 606752 > 32⁷³ [i]