MinT - Best Known (209, 253, s)-Nets in Base 4

Best Known (209, 253, s)-Nets in Base 4

(209, 253, 1572)-Net over F₄ — Constructive and digital

Digital (209, 253, 1572)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 37, 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 (172, 216, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 72, 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]

(209, 253, 19641)-Net over F₄ — Digital

Digital (209, 253, 19641)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(209, 253, large)-Net in Base 4 — Upper bound on s

There is no (209, 253, large)-net in base 4, because

42 times m-reduction [i] would yield (209, 211, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]