MinT - Best Known (52−34, 52, s)-Nets in Base 256

Best Known (52−34, 52, s)-Nets in Base 256

(52−34, 52, 515)-Net over F₂₅₆ — Constructive and digital

Digital (18, 52, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 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, 35, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(52−34, 52, 546)-Net over F₂₅₆ — Digital

Digital (18, 52, 546)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 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, 35, 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]

(52−34, 52, 654305)-Net in Base 256 — Upper bound on s

There is no (18, 52, 654306)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 169231 389836 346843 470606 328617 128070 029866 731584 961076 760174 681586 459830 131340 141568 336642 593950 087957 480347 423328 148512 627286 > 256⁵² [i]