MinT - Best Known (256−115, 256, s)-Nets in Base 4

Best Known (256−115, 256, s)-Nets in Base 4

(256−115, 256, 131)-Net over F₄ — Constructive and digital

Digital (141, 256, 131)-net over F₄, using

1 times m-reduction [i] based on digital (141, 257, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 68, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 189, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(256−115, 256, 251)-Net over F₄ — Digital

Digital (141, 256, 251)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−115, 256, 3586)-Net in Base 4 — Upper bound on s

There is no (141, 256, 3587)-net in base 4, because

1 times m-reduction [i] would yield (141, 255, 3587)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3377 039077 704014 617294 561278 505291 245491 364362 302933 597652 933268 782766 113349 516150 547676 229493 594740 605693 859062 299405 020219 557687 506241 231068 076142 217600 > 4²⁵⁵ [i]