MinT - Best Known (211−71, 211, s)-Nets in Base 4

Best Known (211−71, 211, s)-Nets in Base 4

(211−71, 211, 164)-Net over F₄ — Constructive and digital

Digital (140, 211, 164)-net over F₄, using

4¹ times duplication [i] based on digital (139, 210, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 56, 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 (83, 154, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 77, 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]

(211−71, 211, 240)-Net in Base 4 — Constructive

(140, 211, 240)-net in base 4, using

3 times m-reduction [i] based on (140, 214, 240)-net in base 4, using
- trace code for nets [i] based on (33, 107, 120)-net in base 16, using
  - 3 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]

(211−71, 211, 557)-Net over F₄ — Digital

Digital (140, 211, 557)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−71, 211, 18960)-Net in Base 4 — Upper bound on s

There is no (140, 211, 18961)-net in base 4, because

1 times m-reduction [i] would yield (140, 210, 18961)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 710846 533224 888276 354638 040395 000629 702038 048541 139081 771883 977576 196931 533887 322640 858103 156565 735010 060863 707626 697416 327424 > 4²¹⁰ [i]