MinT - Best Known (106, 159, s)-Nets in Base 4

Best Known (106, 159, s)-Nets in Base 4

(106, 159, 195)-Net over F₄ — Constructive and digital

Digital (106, 159, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 53, 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]

(106, 159, 208)-Net in Base 4 — Constructive

(106, 159, 208)-net in base 4, using

t-expansion [i] based on (105, 159, 208)-net in base 4, using
- 1 times m-reduction [i] based on (105, 160, 208)-net in base 4, using
  - trace code for nets [i] based on (25, 80, 104)-net in base 16, using
    - base change [i] based on digital (9, 64, 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]

(106, 159, 452)-Net over F₄ — Digital

Digital (106, 159, 452)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(106, 159, 16005)-Net in Base 4 — Upper bound on s

There is no (106, 159, 16006)-net in base 4, because

1 times m-reduction [i] would yield (106, 158, 16006)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 133598 504921 718685 320670 106466 727538 797251 114342 404915 681889 187100 998600 384225 175757 640695 659296 > 4¹⁵⁸ [i]