MinT - Best Known (77, 77+45, s)-Nets in Base 4

Best Known (77, 77+45, s)-Nets in Base 4

(77, 77+45, 130)-Net over F₄ — Constructive and digital

Digital (77, 122, 130)-net over F₄, using

20 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+45, 253)-Net over F₄ — Digital

Digital (77, 122, 253)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+45, 6163)-Net in Base 4 — Upper bound on s

There is no (77, 122, 6164)-net in base 4, because

1 times m-reduction [i] would yield (77, 121, 6164)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 086809 863000 616297 598177 582398 745689 801068 743565 270380 049897 047975 769600 > 4¹²¹ [i]