MinT - Best Known (172, 172+35, s)-Nets in Base 4

Best Known (172, 172+35, s)-Nets in Base 4

(172, 172+35, 1548)-Net over F₄ — Constructive and digital

Digital (172, 207, 1548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (154, 189, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 63, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(172, 172+35, 20902)-Net over F₄ — Digital

Digital (172, 207, 20902)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+35, large)-Net in Base 4 — Upper bound on s

There is no (172, 207, large)-net in base 4, because

33 times m-reduction [i] would yield (172, 174, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]