MinT - Best Known (176, 237, s)-Nets in Base 4

Best Known (176, 237, s)-Nets in Base 4

(176, 237, 545)-Net over F₄ — Constructive and digital

Digital (176, 237, 545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 33, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (143, 204, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 68, 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]

(176, 237, 648)-Net in Base 4 — Constructive

(176, 237, 648)-net in base 4, using

t-expansion [i] based on (175, 237, 648)-net in base 4, using
- trace code for nets [i] based on (17, 79, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(176, 237, 1876)-Net over F₄ — Digital

Digital (176, 237, 1876)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(176, 237, 218687)-Net in Base 4 — Upper bound on s

There is no (176, 237, 218688)-net in base 4, because

1 times m-reduction [i] would yield (176, 236, 218688)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12195 107941 322839 738238 093528 534641 482486 842410 881363 082903 235849 651259 289525 041614 864604 633188 197081 922439 498980 055677 659689 659557 564710 188605 > 4²³⁶ [i]