MinT - Best Known (77, 77+63, s)-Nets in Base 4

Best Known (77, 77+63, s)-Nets in Base 4

(77, 77+63, 130)-Net over F₄ — Constructive and digital

Digital (77, 140, 130)-net over F₄, using

2 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+63, 148)-Net over F₄ — Digital

Digital (77, 140, 148)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+63, 2047)-Net in Base 4 — Upper bound on s

There is no (77, 140, 2048)-net in base 4, because

1 times m-reduction [i] would yield (77, 139, 2048)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 491713 949197 707375 429726 285894 107179 814378 350516 618665 488589 266112 047187 625338 033025 > 4¹³⁹ [i]