MinT - Best Known (216−72, 216, s)-Nets in Base 4

Best Known (216−72, 216, s)-Nets in Base 4

(216−72, 216, 195)-Net over F₄ — Constructive and digital

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]

(216−72, 216, 240)-Net in Base 4 — Constructive

(144, 216, 240)-net in base 4, using

t-expansion [i] based on (143, 216, 240)-net in base 4, using
- 4 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]

(216−72, 216, 588)-Net over F₄ — Digital

Digital (144, 216, 588)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−72, 216, 19467)-Net in Base 4 — Upper bound on s

There is no (144, 216, 19468)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11093 222954 145064 382870 412281 339707 237432 556964 882486 388233 716636 742532 759442 341853 370474 858496 063421 321392 819141 889709 890356 056640 > 4²¹⁶ [i]