MinT - Best Known (212−72, 212, s)-Nets in Base 4

Best Known (212−72, 212, s)-Nets in Base 4

(212−72, 212, 163)-Net over F₄ — Constructive and digital

Digital (140, 212, 163)-net over F₄, using

t-expansion [i] based on digital (139, 212, 163)-net over F₄, using
- 2 times m-reduction [i] based on digital (139, 214, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 52, 33)-net over F₄, using
      - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
    2. digital (87, 162, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 81, 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]

(212−72, 212, 240)-Net in Base 4 — Constructive

(140, 212, 240)-net in base 4, using

2 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]

(212−72, 212, 541)-Net over F₄ — Digital

Digital (140, 212, 541)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−72, 212, 16684)-Net in Base 4 — Upper bound on s

There is no (140, 212, 16685)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 368397 395187 415768 600753 190765 631791 086260 717429 036247 954375 755529 069775 024324 068264 847216 534735 912733 711490 276625 348273 730795 > 4²¹² [i]