MinT - Best Known (234−78, 234, s)-Nets in Base 4

Best Known (234−78, 234, s)-Nets in Base 4

(234−78, 234, 195)-Net over F₄ — Constructive and digital

Digital (156, 234, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 78, 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]

(234−78, 234, 240)-Net in Base 4 — Constructive

(156, 234, 240)-net in base 4, using

t-expansion [i] based on (155, 234, 240)-net in base 4, using
- 6 times m-reduction [i] based on (155, 240, 240)-net in base 4, using
  - trace code for nets [i] based on (35, 120, 120)-net in base 16, using
    - base change [i] based on digital (11, 96, 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]

(234−78, 234, 631)-Net over F₄ — Digital

Digital (156, 234, 631)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−78, 234, 20989)-Net in Base 4 — Upper bound on s

There is no (156, 234, 20990)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 762 809438 800142 990394 157005 521064 179466 735253 063586 166130 120812 329308 544044 528927 888189 068877 275421 694276 686664 954811 165903 316098 416966 162788 > 4²³⁴ [i]