MinT - Best Known (255−65, 255, s)-Nets in Base 4

Best Known (255−65, 255, s)-Nets in Base 4

(255−65, 255, 552)-Net over F₄ — Constructive and digital

Digital (190, 255, 552)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 39, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (151, 216, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 72, 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]

(255−65, 255, 648)-Net in Base 4 — Constructive

(190, 255, 648)-net in base 4, using

3 times m-reduction [i] based on (190, 258, 648)-net in base 4, using
- trace code for nets [i] based on (18, 86, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
    - base change [i] based on digital (5, 78, 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]

(255−65, 255, 2092)-Net over F₄ — Digital

Digital (190, 255, 2092)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−65, 255, 256192)-Net in Base 4 — Upper bound on s

There is no (190, 255, 256193)-net in base 4, because

1 times m-reduction [i] would yield (190, 254, 256193)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 838 069445 582011 224074 619018 938237 270757 473793 048862 171041 960878 378760 242030 836637 762497 767793 589496 379702 961236 282890 416846 424005 314076 224428 557114 285890 > 4²⁵⁴ [i]