MinT - Best Known (218−78, 218, s)-Nets in Base 4

Best Known (218−78, 218, s)-Nets in Base 4

(218−78, 218, 157)-Net over F₄ — Constructive and digital

Digital (140, 218, 157)-net over F₄, using

1 times m-reduction [i] based on digital (140, 219, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 49, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (91, 170, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 85, 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]

(218−78, 218, 208)-Net in Base 4 — Constructive

(140, 218, 208)-net in base 4, using

trace code for nets [i] based on (31, 109, 104)-net in base 16, using
- 1 times m-reduction [i] based on (31, 110, 104)-net in base 16, using
  - base change [i] based on digital (9, 88, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(218−78, 218, 462)-Net over F₄ — Digital

Digital (140, 218, 462)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−78, 218, 11871)-Net in Base 4 — Upper bound on s

There is no (140, 218, 11872)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 177877 411443 279992 466583 178313 326639 410822 369591 041074 487860 631388 814706 534442 059508 762325 137956 232113 334063 349155 974963 554634 961354 > 4²¹⁸ [i]