MinT - Best Known (219−75, 219, s)-Nets in Base 4

Best Known (219−75, 219, s)-Nets in Base 4

(219−75, 219, 163)-Net over F₄ — Constructive and digital

Digital (144, 219, 163)-net over F₄, using

3 times m-reduction [i] based on digital (144, 222, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 54, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (90, 168, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 84, 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]

(219−75, 219, 240)-Net in Base 4 — Constructive

(144, 219, 240)-net in base 4, using

t-expansion [i] based on (143, 219, 240)-net in base 4, using
- 1 times m-reduction [i] based on (143, 220, 240)-net in base 4, using
  - trace code for nets [i] based on (33, 110, 120)-net in base 16, using
    - base change [i] based on digital (11, 88, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(219−75, 219, 540)-Net over F₄ — Digital

Digital (144, 219, 540)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−75, 219, 17190)-Net in Base 4 — Upper bound on s

There is no (144, 219, 17191)-net in base 4, because

1 times m-reduction [i] would yield (144, 218, 17191)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 177501 097484 040335 118479 578094 670821 524187 536143 492813 236858 064913 208439 489995 304596 661260 267098 721274 794127 102400 074822 782094 326830 > 4²¹⁸ [i]