MinT - Best Known (215−57, 215, s)-Nets in Base 4

Best Known (215−57, 215, s)-Nets in Base 4

(215−57, 215, 531)-Net over F₄ — Constructive and digital

Digital (158, 215, 531)-net over F₄, using

t-expansion [i] based on digital (157, 215, 531)-net over F₄, using
- 10 times m-reduction [i] based on digital (157, 225, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 75, 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]

(215−57, 215, 576)-Net in Base 4 — Constructive

(158, 215, 576)-net in base 4, using

1 times m-reduction [i] based on (158, 216, 576)-net in base 4, using
- trace code for nets [i] based on (14, 72, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
    - base change [i] based on digital (3, 66, 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]

(215−57, 215, 1510)-Net over F₄ — Digital

Digital (158, 215, 1510)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−57, 215, 150409)-Net in Base 4 — Upper bound on s

There is no (158, 215, 150410)-net in base 4, because

1 times m-reduction [i] would yield (158, 214, 150410)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 693 183139 988567 685382 279214 463520 384186 771582 551765 029034 864328 470662 639676 821247 153828 591700 127486 166266 698624 695786 500350 730272 > 4²¹⁴ [i]