MinT - Best Known (216−75, 216, s)-Nets in Base 4

Best Known (216−75, 216, s)-Nets in Base 4

(216−75, 216, 163)-Net over F₄ — Constructive and digital

Digital (141, 216, 163)-net over F₄, using

1 times m-reduction [i] based on digital (141, 217, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 53, 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 (88, 164, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 82, 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]

(216−75, 216, 240)-Net in Base 4 — Constructive

(141, 216, 240)-net in base 4, using

trace code for nets [i] based on (33, 108, 120)-net in base 16, using
- 2 times m-reduction [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]

(216−75, 216, 508)-Net over F₄ — Digital

Digital (141, 216, 508)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−75, 216, 15359)-Net in Base 4 — Upper bound on s

There is no (141, 216, 15360)-net in base 4, because

1 times m-reduction [i] would yield (141, 215, 15360)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2772 698174 948094 955419 917471 498925 772768 700724 658995 791881 673717 671750 450487 748571 517302 740873 060815 045386 633404 767609 639781 443425 > 4²¹⁵ [i]