MinT - Best Known (259−44, 259, s)-Nets in Base 4

Best Known (259−44, 259, s)-Nets in Base 4

(259−44, 259, 1574)-Net over F₄ — Constructive and digital

Digital (215, 259, 1574)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 43, 35)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (3, 14, 14)-net over F₄, using
      - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
        an explicitly constructive algebraic function field [i]
    2. digital (7, 29, 21)-net over F₄, using
      - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, 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]

(259−44, 259, 23828)-Net over F₄ — Digital

Digital (215, 259, 23828)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−44, 259, large)-Net in Base 4 — Upper bound on s

There is no (215, 259, large)-net in base 4, because

42 times m-reduction [i] would yield (215, 217, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]