MinT - Best Known (167−43, 167, s)-Nets in Base 4

Best Known (167−43, 167, s)-Nets in Base 4

(167−43, 167, 531)-Net over F₄ — Constructive and digital

Digital (124, 167, 531)-net over F₄, using

t-expansion [i] based on digital (123, 167, 531)-net over F₄, using
- 7 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]

(167−43, 167, 576)-Net in Base 4 — Constructive

(124, 167, 576)-net in base 4, using

t-expansion [i] based on (123, 167, 576)-net in base 4, using
- 1 times m-reduction [i] based on (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]

(167−43, 167, 1384)-Net over F₄ — Digital

Digital (124, 167, 1384)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−43, 167, 166137)-Net in Base 4 — Upper bound on s

There is no (124, 167, 166138)-net in base 4, because

1 times m-reduction [i] would yield (124, 166, 166138)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8749 517633 138700 984024 458809 462657 706667 579295 443382 199732 369236 566509 518850 354467 140598 179825 059290 > 4¹⁶⁶ [i]