MinT - Best Known (77, 77+65, s)-Nets in Base 4

Best Known (77, 77+65, s)-Nets in Base 4

(77, 77+65, 130)-Net over F₄ — Constructive and digital

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+65, 141)-Net over F₄ — Digital

Digital (77, 142, 141)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+65, 1890)-Net in Base 4 — Upper bound on s

There is no (77, 142, 1891)-net in base 4, because

1 times m-reduction [i] would yield (77, 141, 1891)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 795367 441224 504780 488518 565739 841619 877876 717352 803743 902742 790026 147588 815673 891367 > 4¹⁴¹ [i]