MinT - Best Known (68−45, 68, s)-Nets in Base 256

Best Known (68−45, 68, s)-Nets in Base 256

(68−45, 68, 515)-Net over F₂₅₆ — Constructive and digital

Digital (23, 68, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 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, 46, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(68−45, 68, 546)-Net over F₂₅₆ — Digital

Digital (23, 68, 546)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 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, 46, 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]

(68−45, 68, 766464)-Net in Base 256 — Upper bound on s

There is no (23, 68, 766465)-net in base 256, because

1 times m-reduction [i] would yield (23, 67, 766465)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 224947 181107 401401 803864 634087 829701 636222 310219 112262 660482 318068 418234 046776 966079 346183 190202 405553 505697 335086 362120 304103 445759 090583 799295 315987 535708 779776 > 256⁶⁷ [i]