MinT - Best Known (144, 144+71, s)-Nets in Base 4

Best Known (144, 144+71, s)-Nets in Base 4

(144, 144+71, 195)-Net over F₄ — Constructive and digital

Digital (144, 215, 195)-net over F₄, using

1 times m-reduction [i] based on digital (144, 216, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 72, 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]

(144, 144+71, 240)-Net in Base 4 — Constructive

(144, 215, 240)-net in base 4, using

t-expansion [i] based on (143, 215, 240)-net in base 4, using
- 5 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]

(144, 144+71, 606)-Net over F₄ — Digital

Digital (144, 215, 606)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(144, 144+71, 22220)-Net in Base 4 — Upper bound on s

There is no (144, 215, 22221)-net in base 4, because

1 times m-reduction [i] would yield (144, 214, 22221)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 693 943963 571705 134001 730488 293522 620683 097867 397085 966498 344371 555454 290253 360500 889532 591614 649397 015232 932563 602761 750326 092408 > 4²¹⁴ [i]