MinT - Best Known (228−61, 228, s)-Nets in Base 4

Best Known (228−61, 228, s)-Nets in Base 4

(228−61, 228, 531)-Net over F₄ — Constructive and digital

Digital (167, 228, 531)-net over F₄, using

12 times m-reduction [i] based on digital (167, 240, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 80, 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]

(228−61, 228, 576)-Net in Base 4 — Constructive

(167, 228, 576)-net in base 4, using

t-expansion [i] based on (166, 228, 576)-net in base 4, using
- trace code for nets [i] based on (14, 76, 192)-net in base 64, using
  - 1 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]

(228−61, 228, 1530)-Net over F₄ — Digital

Digital (167, 228, 1530)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−61, 228, 144271)-Net in Base 4 — Upper bound on s

There is no (167, 228, 144272)-net in base 4, because

1 times m-reduction [i] would yield (167, 227, 144272)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46521 245830 366781 662447 843632 815705 208951 150850 913605 888859 958849 186576 020144 660531 961268 209945 685910 189821 379121 719515 313939 171414 887144 > 4²²⁷ [i]