MinT - Best Known (166−28, 166, s)-Nets in Base 4

Best Known (166−28, 166, s)-Nets in Base 4

(166−28, 166, 1542)-Net over F₄ — Constructive and digital

Digital (138, 166, 1542)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (40, 54, 514)-net over F₄, using
  - trace code for nets [i] based on digital (13, 27, 257)-net over F₁₆, using
    - base reduction for projective spaces (embedding PG(13,256) in PG(26,16)) for nets [i] based on digital (0, 14, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. digital (84, 112, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 28, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(166−28, 166, 18330)-Net over F₄ — Digital

Digital (138, 166, 18330)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−28, 166, large)-Net in Base 4 — Upper bound on s

There is no (138, 166, large)-net in base 4, because

26 times m-reduction [i] would yield (138, 140, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]