MinT - Best Known (238−40, 238, s)-Nets in Base 4

Best Known (238−40, 238, s)-Nets in Base 4

(238−40, 238, 1569)-Net over F₄ — Constructive and digital

Digital (198, 238, 1569)-net over F₄, using

4¹ times duplication [i] based on digital (197, 237, 1569)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 33, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (164, 204, 1539)-net over F₄, using
    - trace code for nets [i] based on digital (28, 68, 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]

(238−40, 238, 24253)-Net over F₄ — Digital

Digital (198, 238, 24253)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−40, 238, large)-Net in Base 4 — Upper bound on s

There is no (198, 238, large)-net in base 4, because

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