MinT - Best Known (131, 131+65, s)-Nets in Base 4

Best Known (131, 131+65, s)-Nets in Base 4

(131, 131+65, 195)-Net over F₄ — Constructive and digital

Digital (131, 196, 195)-net over F₄, using

4¹ times duplication [i] based on digital (130, 195, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 65, 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]

(131, 131+65, 240)-Net in Base 4 — Constructive

(131, 196, 240)-net in base 4, using

4 times m-reduction [i] based on (131, 200, 240)-net in base 4, using
- trace code for nets [i] based on (31, 100, 120)-net in base 16, using
  - base change [i] based on digital (11, 80, 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]

(131, 131+65, 551)-Net over F₄ — Digital

Digital (131, 196, 551)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+65, 19860)-Net in Base 4 — Upper bound on s

There is no (131, 196, 19861)-net in base 4, because

1 times m-reduction [i] would yield (131, 195, 19861)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2523 669014 299893 535892 054045 202858 072017 408309 643593 522473 440938 782875 262311 665291 614675 167685 573504 254144 518131 564060 > 4¹⁹⁵ [i]