MinT - Best Known (47−31, 47, s)-Nets in Base 256

Best Known (47−31, 47, s)-Nets in Base 256

(47−31, 47, 515)-Net over F₂₅₆ — Constructive and digital

Digital (16, 47, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 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, 32, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(47−31, 47, 546)-Net over F₂₅₆ — Digital

Digital (16, 47, 546)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 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, 32, 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]

(47−31, 47, 611632)-Net in Base 256 — Upper bound on s

There is no (16, 47, 611633)-net in base 256, because

1 times m-reduction [i] would yield (16, 46, 611633)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 601 231763 041816 286753 349999 750001 522574 440078 651537 022829 023248 367371 663532 273831 195052 209981 505396 037638 696976 > 256⁴⁶ [i]