MinT - Best Known (168−43, 168, s)-Nets in Base 4

Best Known (168−43, 168, s)-Nets in Base 4

(168−43, 168, 531)-Net over F₄ — Constructive and digital

Digital (125, 168, 531)-net over F₄, using

9 times m-reduction [i] based on digital (125, 177, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 59, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(168−43, 168, 648)-Net in Base 4 — Constructive

(125, 168, 648)-net in base 4, using

trace code for nets [i] based on (13, 56, 216)-net in base 64, using
- base change [i] based on digital (5, 48, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(168−43, 168, 1430)-Net over F₄ — Digital

Digital (125, 168, 1430)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−43, 168, 177476)-Net in Base 4 — Upper bound on s

There is no (125, 168, 177477)-net in base 4, because

1 times m-reduction [i] would yield (125, 167, 177477)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34999 320863 767619 185047 484319 512411 317265 876018 522632 342570 093730 509726 593898 275745 779316 763462 437232 > 4¹⁶⁷ [i]