MinT - Best Known (64−42, 64, s)-Nets in Base 256

Best Known (64−42, 64, s)-Nets in Base 256

(64−42, 64, 515)-Net over F₂₅₆ — Constructive and digital

Digital (22, 64, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 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, 43, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(64−42, 64, 546)-Net over F₂₅₆ — Digital

Digital (22, 64, 546)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 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, 43, 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]

(64−42, 64, 743608)-Net in Base 256 — Upper bound on s

There is no (22, 64, 743609)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 13407 904888 428931 650663 978598 028065 421464 162038 621854 911144 647207 838952 283287 288780 612837 292981 402708 673385 025871 841097 900762 877158 276563 719701 594823 018696 > 256⁶⁴ [i]