MinT - Best Known (163−41, 163, s)-Nets in Base 4

Best Known (163−41, 163, s)-Nets in Base 4

(163−41, 163, 531)-Net over F₄ — Constructive and digital

Digital (122, 163, 531)-net over F₄, using

t-expansion [i] based on digital (121, 163, 531)-net over F₄, using
- 8 times m-reduction [i] based on digital (121, 171, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 57, 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]

(163−41, 163, 648)-Net in Base 4 — Constructive

(122, 163, 648)-net in base 4, using

4¹ times duplication [i] based on (121, 162, 648)-net in base 4, using
- trace code for nets [i] based on (13, 54, 216)-net in base 64, using
  - 2 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]

(163−41, 163, 1513)-Net over F₄ — Digital

Digital (122, 163, 1513)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−41, 163, 208370)-Net in Base 4 — Upper bound on s

There is no (122, 163, 208371)-net in base 4, because

1 times m-reduction [i] would yield (122, 162, 208371)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34 175891 618291 174859 701264 342977 034262 853226 361939 533778 628366 689858 269696 562939 562721 818719 404296 > 4¹⁶² [i]