MinT - Best Known (141, 212, s)-Nets in Base 4

Best Known (141, 212, s)-Nets in Base 4

(141, 212, 164)-Net over F₄ — Constructive and digital

Digital (141, 212, 164)-net over F₄, using

1 times m-reduction [i] based on digital (141, 213, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 57, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (84, 156, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 78, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(141, 212, 240)-Net in Base 4 — Constructive

(141, 212, 240)-net in base 4, using

4 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]

(141, 212, 569)-Net over F₄ — Digital

Digital (141, 212, 569)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(141, 212, 19727)-Net in Base 4 — Upper bound on s

There is no (141, 212, 19728)-net in base 4, because

1 times m-reduction [i] would yield (141, 211, 19728)-net in base 4, but
- 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]