MinT - Best Known (222−76, 222, s)-Nets in Base 4

Best Known (222−76, 222, s)-Nets in Base 4

(222−76, 222, 163)-Net over F₄ — Constructive and digital

Digital (146, 222, 163)-net over F₄, using

t-expansion [i] based on digital (145, 222, 163)-net over F₄, using
- 2 times m-reduction [i] based on digital (145, 224, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 54, 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 (91, 170, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 85, 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]

(222−76, 222, 240)-Net in Base 4 — Constructive

(146, 222, 240)-net in base 4, using

2 times m-reduction [i] based on (146, 224, 240)-net in base 4, using
- trace code for nets [i] based on (34, 112, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
    - base change [i] based on digital (11, 92, 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]

(222−76, 222, 547)-Net over F₄ — Digital

Digital (146, 222, 547)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−76, 222, 16449)-Net in Base 4 — Upper bound on s

There is no (146, 222, 16450)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 457625 170638 711517 791934 608039 062594 690685 792935 407880 073189 711183 621886 786731 117873 942897 503434 044875 611456 454473 175838 560077 795360 > 4²²² [i]