MinT - Best Known (111, 111+39, s)-Nets in Base 4

Best Known (111, 111+39, s)-Nets in Base 4

(111, 111+39, 531)-Net over F₄ — Constructive and digital

Digital (111, 150, 531)-net over F₄, using

6 times m-reduction [i] based on digital (111, 156, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 52, 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]

(111, 111+39, 576)-Net in Base 4 — Constructive

(111, 150, 576)-net in base 4, using

4³ 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]

(111, 111+39, 1211)-Net over F₄ — Digital

Digital (111, 150, 1211)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+39, 139143)-Net in Base 4 — Upper bound on s

There is no (111, 150, 139144)-net in base 4, because

1 times m-reduction [i] would yield (111, 149, 139144)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 509264 766496 689099 594370 780171 742026 367220 051247 411607 834016 446791 936861 641695 822907 943048 > 4¹⁴⁹ [i]