MinT - Best Known (77, 77+51, s)-Nets in Base 4

Best Known (77, 77+51, s)-Nets in Base 4

(77, 77+51, 130)-Net over F₄ — Constructive and digital

Digital (77, 128, 130)-net over F₄, using

14 times m-reduction [i] based on digital (77, 142, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 71, 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]

(77, 77+51, 203)-Net over F₄ — Digital

Digital (77, 128, 203)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+51, 3860)-Net in Base 4 — Upper bound on s

There is no (77, 128, 3861)-net in base 4, because

1 times m-reduction [i] would yield (77, 127, 3861)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 28959 744085 738266 554222 913407 211787 423330 501723 600851 428323 195318 745085 458176 > 4¹²⁷ [i]