MinT - Best Known (230−61, 230, s)-Nets in Base 4

Best Known (230−61, 230, s)-Nets in Base 4

(230−61, 230, 531)-Net over F₄ — Constructive and digital

Digital (169, 230, 531)-net over F₄, using

13 times m-reduction [i] based on digital (169, 243, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 81, 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]

(230−61, 230, 576)-Net in Base 4 — Constructive

(169, 230, 576)-net in base 4, using

t-expansion [i] based on (168, 230, 576)-net in base 4, using
- 1 times m-reduction [i] based on (168, 231, 576)-net in base 4, using
  - trace code for nets [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]

(230−61, 230, 1601)-Net over F₄ — Digital

Digital (169, 230, 1601)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−61, 230, 158243)-Net in Base 4 — Upper bound on s

There is no (169, 230, 158244)-net in base 4, because

1 times m-reduction [i] would yield (169, 229, 158244)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 744409 721986 270518 891367 007245 681582 039793 031740 366486 897854 679130 332805 058241 639249 001963 408242 151439 947920 371528 361386 461007 151302 595608 > 4²²⁹ [i]