MinT - Best Known (19, 19+37, s)-Nets in Base 256

Best Known (19, 19+37, s)-Nets in Base 256

(19, 19+37, 515)-Net over F₂₅₆ — Constructive and digital

Digital (19, 56, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 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, 38, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(19, 19+37, 546)-Net over F₂₅₆ — Digital

Digital (19, 56, 546)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 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, 38, 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]

(19, 19+37, 676231)-Net in Base 256 — Upper bound on s

There is no (19, 56, 676232)-net in base 256, because

1 times m-reduction [i] would yield (19, 55, 676232)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 839253 897984 667759 283608 836113 390269 292617 891683 183997 412174 980647 279406 464815 558352 774932 652398 931576 710197 122669 670479 099697 186856 > 256⁵⁵ [i]