MinT - Best Known (168−56, 168, s)-Nets in Base 4

Best Known (168−56, 168, s)-Nets in Base 4

(168−56, 168, 195)-Net over F₄ — Constructive and digital

Digital (112, 168, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 56, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(168−56, 168, 240)-Net in Base 4 — Constructive

(112, 168, 240)-net in base 4, using

trace code for nets [i] based on (28, 84, 120)-net in base 16, using
- 1 times m-reduction [i] based on (28, 85, 120)-net in base 16, using
  - base change [i] based on digital (11, 68, 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]

(168−56, 168, 473)-Net over F₄ — Digital

Digital (112, 168, 473)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−56, 168, 15402)-Net in Base 4 — Upper bound on s

There is no (112, 168, 15403)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 140025 083012 789310 546796 489925 370363 761526 363804 098879 324390 922264 656784 157739 393689 171672 554287 380120 > 4¹⁶⁸ [i]