MinT - Best Known (168, 168+41, s)-Nets in Base 4

Best Known (168, 168+41, s)-Nets in Base 4

(168, 168+41, 1539)-Net over F₄ — Constructive and digital

Digital (168, 209, 1539)-net over F₄, using

1 times m-reduction [i] based on digital (168, 210, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 70, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(168, 168+41, 7372)-Net over F₄ — Digital

Digital (168, 209, 7372)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+41, 5053677)-Net in Base 4 — Upper bound on s

There is no (168, 209, 5053678)-net in base 4, because

1 times m-reduction [i] would yield (168, 208, 5053678)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169230 876161 700316 582983 924589 300335 954899 616067 665676 894194 432899 869348 556100 371540 001030 530674 160213 893010 200755 822877 746226 > 4²⁰⁸ [i]