MinT - Best Known (123, 123+45, s)-Nets in Base 4

Best Known (123, 123+45, s)-Nets in Base 4

(123, 123+45, 531)-Net over F₄ — Constructive and digital

Digital (123, 168, 531)-net over F₄, using

6 times m-reduction [i] based on digital (123, 174, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 58, 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]

(123, 123+45, 576)-Net in Base 4 — Constructive

(123, 168, 576)-net in base 4, using

trace code for nets [i] based on (11, 56, 192)-net in base 64, using
- base change [i] based on digital (3, 48, 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]

(123, 123+45, 1166)-Net over F₄ — Digital

Digital (123, 168, 1166)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+45, 112161)-Net in Base 4 — Upper bound on s

There is no (123, 168, 112162)-net in base 4, because

1 times m-reduction [i] would yield (123, 167, 112162)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34996 636595 037579 164890 015047 353349 782324 563331 059467 150672 520725 932437 956344 566428 087366 774985 096720 > 4¹⁶⁷ [i]