MinT - Best Known (239−41, 239, s)-Nets in Base 4

Best Known (239−41, 239, s)-Nets in Base 4

(239−41, 239, 1567)-Net over F₄ — Constructive and digital

Digital (198, 239, 1567)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 32, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (166, 207, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 69, 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]

(239−41, 239, 20816)-Net over F₄ — Digital

Digital (198, 239, 20816)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−41, 239, large)-Net in Base 4 — Upper bound on s

There is no (198, 239, large)-net in base 4, because

39 times m-reduction [i] would yield (198, 200, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]