MinT - Best Known (86, 86+69, s)-Nets in Base 4

Best Known (86, 86+69, s)-Nets in Base 4

(86, 86+69, 130)-Net over F₄ — Constructive and digital

Digital (86, 155, 130)-net over F₄, using

5 times m-reduction [i] based on digital (86, 160, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 80, 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]

(86, 86+69, 166)-Net over F₄ — Digital

Digital (86, 155, 166)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+69, 2378)-Net in Base 4 — Upper bound on s

There is no (86, 155, 2379)-net in base 4, because

1 times m-reduction [i] would yield (86, 154, 2379)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 525 253575 732954 311638 664636 482353 157409 564532 717456 112263 621423 631695 695995 743223 789339 846174 > 4¹⁵⁴ [i]