MinT - Best Known (164−69, 164, s)-Nets in Base 4

Best Known (164−69, 164, s)-Nets in Base 4

(164−69, 164, 130)-Net over F₄ — Constructive and digital

Digital (95, 164, 130)-net over F₄, using

14 times m-reduction [i] based on digital (95, 178, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 89, 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]

(164−69, 164, 209)-Net over F₄ — Digital

Digital (95, 164, 209)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−69, 164, 3445)-Net in Base 4 — Upper bound on s

There is no (95, 164, 3446)-net in base 4, because

1 times m-reduction [i] would yield (95, 163, 3446)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 137 752939 427278 266494 656035 101305 647696 084059 102636 162246 536192 527274 563129 309857 213199 405938 221690 > 4¹⁶³ [i]