MinT - Best Known (110, 110+39, s)-Nets in Base 4

Best Known (110, 110+39, s)-Nets in Base 4

(110, 110+39, 531)-Net over F₄ — Constructive and digital

Digital (110, 149, 531)-net over F₄, using

t-expansion [i] based on digital (109, 149, 531)-net over F₄, using
- 4 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+39, 576)-Net in Base 4 — Constructive

(110, 149, 576)-net in base 4, using

4² times duplication [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]

(110, 110+39, 1168)-Net over F₄ — Digital

Digital (110, 149, 1168)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(110, 110+39, 129352)-Net in Base 4 — Upper bound on s

There is no (110, 149, 129353)-net in base 4, because

1 times m-reduction [i] would yield (110, 148, 129353)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 127332 965283 987891 365824 758044 013538 570735 850904 075730 719449 516217 282960 166906 408795 119904 > 4¹⁴⁸ [i]