MinT - Best Known (64, 64+42, s)-Nets in Base 16

Best Known (64, 64+42, s)-Nets in Base 16

(64, 64+42, 538)-Net over F₁₆ — Constructive and digital

Digital (64, 106, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (42, 84, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 42, 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]

(64, 64+42, 1417)-Net over F₁₆ — Digital

Digital (64, 106, 1417)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+42, 692361)-Net in Base 16 — Upper bound on s

There is no (64, 106, 692362)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 43 323186 615180 988602 031358 125508 817015 155344 605309 405764 111191 358462 231505 547284 451748 165658 033310 287603 858732 113024 953590 812656 > 16¹⁰⁶ [i]