MinT - Best Known (179−47, 179, s)-Nets in Base 4

Best Known (179−47, 179, s)-Nets in Base 4

(179−47, 179, 531)-Net over F₄ — Constructive and digital

Digital (132, 179, 531)-net over F₄, using

t-expansion [i] based on digital (131, 179, 531)-net over F₄, using
- 7 times m-reduction [i] based on digital (131, 186, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 62, 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]

(179−47, 179, 576)-Net in Base 4 — Constructive

(132, 179, 576)-net in base 4, using

1 times m-reduction [i] based on (132, 180, 576)-net in base 4, using
- trace code for nets [i] based on (12, 60, 192)-net in base 64, using
  - 3 times m-reduction [i] based on (12, 63, 192)-net in base 64, using
    - base change [i] based on digital (3, 54, 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]

(179−47, 179, 1344)-Net over F₄ — Digital

Digital (132, 179, 1344)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(179−47, 179, 143449)-Net in Base 4 — Upper bound on s

There is no (132, 179, 143450)-net in base 4, because

1 times m-reduction [i] would yield (132, 178, 143450)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 146793 489752 181090 246694 966474 410780 773136 070872 101720 282999 246014 765440 409113 706102 114102 792716 745006 484836 > 4¹⁷⁸ [i]