MinT - Best Known (109, 170, s)-Nets in Base 4

Best Known (109, 170, s)-Nets in Base 4

(109, 170, 147)-Net over F₄ — Constructive and digital

Digital (109, 170, 147)-net over F₄, using

4¹ times duplication [i] based on digital (108, 169, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 35, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 134, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 67, 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]

(109, 170, 196)-Net in Base 4 — Constructive

(109, 170, 196)-net in base 4, using

trace code for nets [i] based on (24, 85, 98)-net in base 16, using
- base change [i] based on digital (7, 68, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(109, 170, 368)-Net over F₄ — Digital

Digital (109, 170, 368)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 170, 9867)-Net in Base 4 — Upper bound on s

There is no (109, 170, 9868)-net in base 4, because

1 times m-reduction [i] would yield (109, 169, 9868)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 561214 196649 926368 360156 371744 639997 785847 225059 071539 089056 724750 277834 589571 940555 403841 397836 602848 > 4¹⁶⁹ [i]