MinT - Best Known (246−67, 246, s)-Nets in Base 4

Best Known (246−67, 246, s)-Nets in Base 4

(246−67, 246, 531)-Net over F₄ — Constructive and digital

Digital (179, 246, 531)-net over F₄, using

12 times m-reduction [i] based on digital (179, 258, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 86, 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]

(246−67, 246, 576)-Net in Base 4 — Constructive

(179, 246, 576)-net in base 4, using

trace code for nets [i] based on (15, 82, 192)-net in base 64, using
- 2 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
  - base change [i] based on digital (3, 72, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(246−67, 246, 1504)-Net over F₄ — Digital

Digital (179, 246, 1504)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−67, 246, 129414)-Net in Base 4 — Upper bound on s

There is no (179, 246, 129415)-net in base 4, because

1 times m-reduction [i] would yield (179, 245, 129415)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3197 404341 571926 456873 441434 254734 666533 862615 371697 828297 495515 117408 948883 039977 201866 459609 962897 169916 660293 889470 915741 117715 065214 710999 033720 > 4²⁴⁵ [i]