MinT - Best Known (217−75, 217, s)-Nets in Base 4

Best Known (217−75, 217, s)-Nets in Base 4

(217−75, 217, 163)-Net over F₄ — Constructive and digital

Digital (142, 217, 163)-net over F₄, using

2 times m-reduction [i] based on digital (142, 219, 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 (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]

(217−75, 217, 240)-Net in Base 4 — Constructive

(142, 217, 240)-net in base 4, using

1 times m-reduction [i] based on (142, 218, 240)-net in base 4, using
- trace code for nets [i] based on (33, 109, 120)-net in base 16, using
  - 1 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]

(217−75, 217, 519)-Net over F₄ — Digital

Digital (142, 217, 519)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−75, 217, 15947)-Net in Base 4 — Upper bound on s

There is no (142, 217, 15948)-net in base 4, because

1 times m-reduction [i] would yield (142, 216, 15948)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11101 764358 510497 361140 710840 766061 312189 564419 726624 295754 457297 433891 319347 162305 667599 903604 857565 197858 305732 295197 358853 374700 > 4²¹⁶ [i]