MinT - Best Known (85, 85+53, s)-Nets in Base 4

Best Known (85, 85+53, s)-Nets in Base 4

(85, 85+53, 130)-Net over F₄ — Constructive and digital

Digital (85, 138, 130)-net over F₄, using

20 times m-reduction [i] based on digital (85, 158, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 79, 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]

(85, 85+53, 245)-Net over F₄ — Digital

Digital (85, 138, 245)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+53, 5209)-Net in Base 4 — Upper bound on s

There is no (85, 138, 5210)-net in base 4, because

1 times m-reduction [i] would yield (85, 137, 5210)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 30410 602292 174027 086996 274496 009941 125173 797836 342848 033813 639740 414582 402502 656968 > 4¹³⁷ [i]