MinT - Best Known (77, 77+57, s)-Nets in Base 4

Best Known (77, 77+57, s)-Nets in Base 4

(77, 77+57, 130)-Net over F₄ — Constructive and digital

Digital (77, 134, 130)-net over F₄, using

8 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+57, 170)-Net over F₄ — Digital

Digital (77, 134, 170)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+57, 2704)-Net in Base 4 — Upper bound on s

There is no (77, 134, 2705)-net in base 4, because

1 times m-reduction [i] would yield (77, 133, 2705)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 119 626438 623160 680849 487823 300525 553996 550440 931521 224207 111988 058852 809018 284128 > 4¹³³ [i]