MinT - Best Known (218−74, 218, s)-Nets in Base 4

Best Known (218−74, 218, s)-Nets in Base 4

(218−74, 218, 164)-Net over F₄ — Constructive and digital

Digital (144, 218, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 58, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (86, 160, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 80, 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]

(218−74, 218, 240)-Net in Base 4 — Constructive

(144, 218, 240)-net in base 4, using

t-expansion [i] based on (143, 218, 240)-net in base 4, using
- 2 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]

(218−74, 218, 555)-Net over F₄ — Digital

Digital (144, 218, 555)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−74, 218, 17190)-Net in Base 4 — Upper bound on s

There is no (144, 218, 17191)-net in base 4, because

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]