MinT - Best Known (136, 136+79, s)-Nets in Base 4

Best Known (136, 136+79, s)-Nets in Base 4

(136, 136+79, 147)-Net over F₄ — Constructive and digital

Digital (136, 215, 147)-net over F₄, using

4¹ times duplication [i] based on digital (135, 214, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 44, 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 (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]

(136, 136+79, 152)-Net in Base 4 — Constructive

(136, 215, 152)-net in base 4, using

3 times m-reduction [i] based on (136, 218, 152)-net in base 4, using
- trace code for nets [i] based on (27, 109, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (27, 110, 76)-net in base 16, using
    - base change [i] based on digital (5, 88, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(136, 136+79, 416)-Net over F₄ — Digital

Digital (136, 215, 416)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+79, 10293)-Net in Base 4 — Upper bound on s

There is no (136, 215, 10294)-net in base 4, because

1 times m-reduction [i] would yield (136, 214, 10294)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 694 119577 451891 475299 310247 187152 268681 642765 199170 432978 653407 051750 992043 186046 273645 226952 080162 610979 622297 356897 233297 329800 > 4²¹⁴ [i]