MinT - Best Known (111, 170, s)-Nets in Base 4

Best Known (111, 170, s)-Nets in Base 4

(111, 170, 157)-Net over F₄ — Constructive and digital

Digital (111, 170, 157)-net over F₄, using

4¹ times duplication [i] based on digital (110, 169, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 39, 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 (71, 130, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 65, 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]

(111, 170, 208)-Net in Base 4 — Constructive

(111, 170, 208)-net in base 4, using

trace code for nets [i] based on (26, 85, 104)-net in base 16, using
- base change [i] based on digital (9, 68, 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]

(111, 170, 413)-Net over F₄ — Digital

Digital (111, 170, 413)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 170, 12523)-Net in Base 4 — Upper bound on s

There is no (111, 170, 12524)-net in base 4, because

1 times m-reduction [i] would yield (111, 169, 12524)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 560127 075654 265960 689791 101530 580101 051914 269264 513483 386324 573711 930870 816544 954472 453952 727301 484616 > 4¹⁶⁹ [i]