MinT - Best Known (78−29, 78, s)-Nets in Base 16

Best Known (78−29, 78, s)-Nets in Base 16

(78−29, 78, 579)-Net over F₁₆ — Constructive and digital

Digital (49, 78, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 20, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (29, 58, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 29, 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]

(78−29, 78, 1717)-Net over F₁₆ — Digital

Digital (49, 78, 1717)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−29, 78, 1690536)-Net in Base 16 — Upper bound on s

There is no (49, 78, 1690537)-net in base 16, because

1 times m-reduction [i] would yield (49, 77, 1690537)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 521 484656 452887 251070 690356 165107 091467 260316 001565 477291 577920 817096 610701 832313 420215 155696 > 16⁷⁷ [i]