MinT - Best Known (72, 72+46, s)-Nets in Base 16

Best Known (72, 72+46, s)-Nets in Base 16

(72, 72+46, 552)-Net over F₁₆ — Constructive and digital

Digital (72, 118, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 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 (46, 92, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 46, 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]

(72, 72+46, 1711)-Net over F₁₆ — Digital

Digital (72, 118, 1711)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+46, 946277)-Net in Base 16 — Upper bound on s

There is no (72, 118, 946278)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 12194 610252 522092 498482 866860 805755 471936 824422 565387 850973 464799 930879 840658 609027 299142 337006 063018 620583 072733 134247 735725 721940 077537 664336 > 16¹¹⁸ [i]