MinT - Best Known (166−42, 166, s)-Nets in Base 4

Best Known (166−42, 166, s)-Nets in Base 4

(166−42, 166, 531)-Net over F₄ — Constructive and digital

Digital (124, 166, 531)-net over F₄, using

t-expansion [i] based on digital (123, 166, 531)-net over F₄, using
- 8 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]

(166−42, 166, 648)-Net in Base 4 — Constructive

(124, 166, 648)-net in base 4, using

4¹ times duplication [i] based on (123, 165, 648)-net in base 4, using
- trace code for nets [i] based on (13, 55, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (13, 56, 216)-net in base 64, using
    - base change [i] based on digital (5, 48, 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]

(166−42, 166, 1494)-Net over F₄ — Digital

Digital (124, 166, 1494)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−42, 166, 166137)-Net in Base 4 — Upper bound on s

There is no (124, 166, 166138)-net in base 4, because

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]