MinT - Best Known (109, 109+37, s)-Nets in Base 4

Best Known (109, 109+37, s)-Nets in Base 4

(109, 109+37, 531)-Net over F₄ — Constructive and digital

Digital (109, 146, 531)-net over F₄, using

7 times m-reduction [i] based on digital (109, 153, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 51, 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]

(109, 109+37, 576)-Net in Base 4 — Constructive

(109, 146, 576)-net in base 4, using

t-expansion [i] based on (108, 146, 576)-net in base 4, using
- 1 times m-reduction [i] based on (108, 147, 576)-net in base 4, using
  - trace code for nets [i] based on (10, 49, 192)-net in base 64, using
    - base change [i] based on digital (3, 42, 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]

(109, 109+37, 1334)-Net over F₄ — Digital

Digital (109, 146, 1334)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+37, 178197)-Net in Base 4 — Upper bound on s

There is no (109, 146, 178198)-net in base 4, because

1 times m-reduction [i] would yield (109, 145, 178198)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1989 429124 889102 671082 585733 912179 038618 782839 108013 395552 371206 111108 524244 881816 926570 > 4¹⁴⁵ [i]