MinT - Best Known (80−31, 80, s)-Nets in Base 16

Best Known (80−31, 80, s)-Nets in Base 16

(80−31, 80, 552)-Net over F₁₆ — Constructive and digital

Digital (49, 80, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (31, 62, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 31, 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]

(80−31, 80, 1320)-Net over F₁₆ — Digital

Digital (49, 80, 1320)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(80−31, 80, 940524)-Net in Base 16 — Upper bound on s

There is no (49, 80, 940525)-net in base 16, because

1 times m-reduction [i] would yield (49, 79, 940525)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 133499 563258 834568 377640 767516 270328 947560 140351 629243 316180 941748 563447 688910 279032 652720 249376 > 16⁷⁹ [i]