MinT - Best Known (143, 143+71, s)-Nets in Base 4

Best Known (143, 143+71, s)-Nets in Base 4

(143, 143+71, 195)-Net over F₄ — Constructive and digital

Digital (143, 214, 195)-net over F₄, using

4¹ times duplication [i] based on digital (142, 213, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 71, 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]

(143, 143+71, 240)-Net in Base 4 — Constructive

(143, 214, 240)-net in base 4, using

6 times m-reduction [i] based on (143, 220, 240)-net in base 4, using
- trace code for nets [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]

(143, 143+71, 593)-Net over F₄ — Digital

Digital (143, 214, 593)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(143, 143+71, 21356)-Net in Base 4 — Upper bound on s

There is no (143, 214, 21357)-net in base 4, because

1 times m-reduction [i] would yield (143, 213, 21357)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 173 499631 055370 632013 881409 546656 899643 284987 851655 859687 286337 588794 820972 434498 445452 585366 416710 892029 698068 464273 204438 626340 > 4²¹³ [i]