MinT - Best Known (194, 194+66, s)-Nets in Base 4

Best Known (194, 194+66, s)-Nets in Base 4

(194, 194+66, 552)-Net over F₄ — Constructive and digital

Digital (194, 260, 552)-net over F₄, using

4¹ times duplication [i] based on digital (193, 259, 552)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 40, 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 (153, 219, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 73, 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]

(194, 194+66, 648)-Net in Base 4 — Constructive

(194, 260, 648)-net in base 4, using

4² times duplication [i] based on (192, 258, 648)-net in base 4, using
- t-expansion [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]

(194, 194+66, 2169)-Net over F₄ — Digital

Digital (194, 260, 2169)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(194, 194+66, 243045)-Net in Base 4 — Upper bound on s

There is no (194, 260, 243046)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 432633 576533 963011 703390 883026 570986 924396 149158 340163 435599 963829 917581 951811 095357 293019 759371 729636 788693 616823 424850 330606 180919 782376 113040 584328 897053 > 4²⁶⁰ [i]