MinT - Best Known (77−30, 77, s)-Nets in Base 16

Best Known (77−30, 77, s)-Nets in Base 16

(77−30, 77, 547)-Net over F₁₆ — Constructive and digital

Digital (47, 77, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (30, 60, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 30, 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]

(77−30, 77, 1240)-Net over F₁₆ — Digital

Digital (47, 77, 1240)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−30, 77, 649859)-Net in Base 16 — Upper bound on s

There is no (47, 77, 649860)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 521 490692 913290 379490 367946 230270 627186 722359 134907 958471 063604 261652 161442 886404 591094 905376 > 16⁷⁷ [i]