MinT - Best Known (78, 78+50, s)-Nets in Base 16

Best Known (78, 78+50, s)-Nets in Base 16

(78, 78+50, 552)-Net over F₁₆ — Constructive and digital

Digital (78, 128, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 28, 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 (50, 100, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 50, 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]

(78, 78+50, 1806)-Net over F₁₆ — Digital

Digital (78, 128, 1806)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+50, 992254)-Net in Base 16 — Upper bound on s

There is no (78, 128, 992255)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 13408 115374 118543 774142 285425 103437 602789 617742 955475 224892 688422 964079 123594 787391 357384 664465 157642 844821 881901 903227 354648 984904 091293 315321 699589 295626 > 16¹²⁸ [i]