MinT - Best Known (66−43, 66, s)-Nets in Base 256

Best Known (66−43, 66, s)-Nets in Base 256

(66−43, 66, 516)-Net over F₂₅₆ — Constructive and digital

Digital (23, 66, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 44, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(66−43, 66, 578)-Net over F₂₅₆ — Digital

Digital (23, 66, 578)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 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]
2. digital (1, 44, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆ (see above)

(66−43, 66, 968331)-Net in Base 256 — Upper bound on s

There is no (23, 66, 968332)-net in base 256, because

1 times m-reduction [i] would yield (23, 65, 968332)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 3 432454 595904 680950 894862 454948 359997 005102 161847 719671 966019 633090 517858 408964 244583 502418 971698 434028 986932 694715 309124 154625 774084 644309 908181 702281 426736 > 256⁶⁵ [i]