MinT - Best Known (210−77, 210, s)-Nets in Base 4

Best Known (210−77, 210, s)-Nets in Base 4

(210−77, 210, 147)-Net over F₄ — Constructive and digital

Digital (133, 210, 147)-net over F₄, using

4¹ times duplication [i] based on digital (132, 209, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 43, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (89, 166, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 83, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(210−77, 210, 196)-Net in Base 4 — Constructive

(133, 210, 196)-net in base 4, using

trace code for nets [i] based on (28, 105, 98)-net in base 16, using
- base change [i] based on digital (7, 84, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(210−77, 210, 410)-Net over F₄ — Digital

Digital (133, 210, 410)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−77, 210, 10225)-Net in Base 4 — Upper bound on s

There is no (133, 210, 10226)-net in base 4, because

1 times m-reduction [i] would yield (133, 209, 10226)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 677751 569588 804762 316926 416382 471708 592913 593212 319927 009472 437119 222186 369579 818536 532589 847422 136182 273615 453989 909569 101944 > 4²⁰⁹ [i]