MinT - Best Known (140, 140+69, s)-Nets in Base 4

Best Known (140, 140+69, s)-Nets in Base 4

(140, 140+69, 195)-Net over F₄ — Constructive and digital

Digital (140, 209, 195)-net over F₄, using

1 times m-reduction [i] based on digital (140, 210, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 70, 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]

(140, 140+69, 240)-Net in Base 4 — Constructive

(140, 209, 240)-net in base 4, using

5 times m-reduction [i] based on (140, 214, 240)-net in base 4, using
- trace code for nets [i] based on (33, 107, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (33, 110, 120)-net in base 16, using
    - base change [i] based on digital (11, 88, 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]

(140, 140+69, 592)-Net over F₄ — Digital

Digital (140, 209, 592)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(140, 140+69, 21726)-Net in Base 4 — Upper bound on s

There is no (140, 209, 21727)-net in base 4, because

1 times m-reduction [i] would yield (140, 208, 21727)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169373 178466 826280 962128 627529 942709 926450 034153 964507 053670 340124 550217 202076 466454 810075 144285 355454 542818 513768 452506 979386 > 4²⁰⁸ [i]