MinT - Best Known (41−27, 41, s)-Nets in Base 256

Best Known (41−27, 41, s)-Nets in Base 256

(41−27, 41, 515)-Net over F₂₅₆ — Constructive and digital

Digital (14, 41, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 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. digital (1, 28, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(41−27, 41, 546)-Net over F₂₅₆ — Digital

Digital (14, 41, 546)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 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. digital (1, 28, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(41−27, 41, 571255)-Net in Base 256 — Upper bound on s

There is no (14, 41, 571256)-net in base 256, because

1 times m-reduction [i] would yield (14, 40, 571256)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 136003 210856 489707 356165 538888 845761 593456 814525 538048 659236 844116 398048 765555 081859 245080 548466 > 256⁴⁰ [i]