MinT - Best Known (241−63, 241, s)-Nets in Base 4

Best Known (241−63, 241, s)-Nets in Base 4

(241−63, 241, 536)-Net over F₄ — Constructive and digital

Digital (178, 241, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 31, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (147, 210, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 70, 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]

(241−63, 241, 648)-Net in Base 4 — Constructive

(178, 241, 648)-net in base 4, using

4¹ times duplication [i] based on (177, 240, 648)-net in base 4, using
- trace code for nets [i] based on (17, 80, 216)-net in base 64, using
  - 4 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(241−63, 241, 1777)-Net over F₄ — Digital

Digital (178, 241, 1777)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−63, 241, 189654)-Net in Base 4 — Upper bound on s

There is no (178, 241, 189655)-net in base 4, because

1 times m-reduction [i] would yield (178, 240, 189655)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 121968 125256 729691 105259 041440 827223 866903 808080 771319 277780 960470 834160 212663 724794 771891 667445 298559 860448 696935 393629 344886 208749 498577 495328 > 4²⁴⁰ [i]