MinT - Best Known (211−70, 211, s)-Nets in Base 4

Best Known (211−70, 211, s)-Nets in Base 4

(211−70, 211, 195)-Net over F₄ — Constructive and digital

Digital (141, 211, 195)-net over F₄, using

4¹ times duplication [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]

(211−70, 211, 240)-Net in Base 4 — Constructive

(141, 211, 240)-net in base 4, using

5 times m-reduction [i] based on (141, 216, 240)-net in base 4, using
- trace code for nets [i] based on (33, 108, 120)-net in base 16, using
  - 2 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]

(211−70, 211, 586)-Net over F₄ — Digital

Digital (141, 211, 586)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−70, 211, 19727)-Net in Base 4 — Upper bound on s

There is no (141, 211, 19728)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 838708 348688 542681 644821 289800 646311 785575 818058 828285 670069 734249 972010 763310 558491 292768 323276 549011 425308 044278 376524 217656 > 4²¹¹ [i]