MinT - Best Known (224−57, 224, s)-Nets in Base 4

Best Known (224−57, 224, s)-Nets in Base 4

(224−57, 224, 546)-Net over F₄ — Constructive and digital

Digital (167, 224, 546)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 32, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (135, 192, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 64, 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]

(224−57, 224, 648)-Net in Base 4 — Constructive

(167, 224, 648)-net in base 4, using

t-expansion [i] based on (166, 224, 648)-net in base 4, using
- 1 times m-reduction [i] based on (166, 225, 648)-net in base 4, using
  - trace code for nets [i] based on (16, 75, 216)-net in base 64, using
    - 2 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
      - base change [i] based on digital (5, 66, 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]

(224−57, 224, 1880)-Net over F₄ — Digital

Digital (167, 224, 1880)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−57, 224, 234865)-Net in Base 4 — Upper bound on s

There is no (167, 224, 234866)-net in base 4, because

1 times m-reduction [i] would yield (167, 223, 234866)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 719129 785433 068276 591394 151862 964174 865068 074237 235381 996689 246227 650457 403337 772576 337953 960530 638563 457760 391097 208578 061073 331704 > 4²²³ [i]