MinT - Best Known (227−78, 227, s)-Nets in Base 4

Best Known (227−78, 227, s)-Nets in Base 4

(227−78, 227, 163)-Net over F₄ — Constructive and digital

Digital (149, 227, 163)-net over F₄, using

t-expansion [i] based on digital (148, 227, 163)-net over F₄, using
- 2 times m-reduction [i] based on digital (148, 229, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 55, 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 (93, 174, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 87, 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]

(227−78, 227, 240)-Net in Base 4 — Constructive

(149, 227, 240)-net in base 4, using

3 times m-reduction [i] based on (149, 230, 240)-net in base 4, using
- trace code for nets [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]

(227−78, 227, 551)-Net over F₄ — Digital

Digital (149, 227, 551)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−78, 227, 16358)-Net in Base 4 — Upper bound on s

There is no (149, 227, 16359)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46529 853969 720161 287981 521131 717075 236496 234573 461188 328745 186068 825203 397846 788559 640324 333385 894966 109020 062370 367254 660063 907539 304660 > 4²²⁷ [i]