MinT - Best Known (110, 110+37, s)-Nets in Base 4

Best Known (110, 110+37, s)-Nets in Base 4

(110, 110+37, 531)-Net over F₄ — Constructive and digital

Digital (110, 147, 531)-net over F₄, using

t-expansion [i] based on digital (109, 147, 531)-net over F₄, using
- 6 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]

(110, 110+37, 648)-Net in Base 4 — Constructive

(110, 147, 648)-net in base 4, using

trace code for nets [i] based on (12, 49, 216)-net in base 64, using
- base change [i] based on digital (5, 42, 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]

(110, 110+37, 1386)-Net over F₄ — Digital

Digital (110, 147, 1386)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(110, 110+37, 192464)-Net in Base 4 — Upper bound on s

There is no (110, 147, 192465)-net in base 4, because

1 times m-reduction [i] would yield (110, 146, 192465)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7957 235666 977815 662055 328674 255927 699419 134784 791044 598942 671102 228024 231098 464537 590640 > 4¹⁴⁶ [i]