MinT - Best Known (67−45, 67, s)-Nets in Base 256

Best Known (67−45, 67, s)-Nets in Base 256

(67−45, 67, 514)-Net over F₂₅₆ — Constructive and digital

Digital (22, 67, 514)-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 (0, 45, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(67−45, 67, 595696)-Net in Base 256 — Upper bound on s

There is no (22, 67, 595697)-net in base 256, because

1 times m-reduction [i] would yield (22, 66, 595697)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 878 711387 895136 661101 084961 350422 852846 977052 234823 845803 662983 952997 191312 480285 663801 130675 731013 020775 341210 755559 908978 449569 666057 347359 964693 049222 092496 > 256⁶⁶ [i]