MinT - Best Known (256−111, 256, s)-Nets in Base 4

Best Known (256−111, 256, s)-Nets in Base 4

(256−111, 256, 137)-Net over F₄ — Constructive and digital

Digital (145, 256, 137)-net over F₄, using

3 times m-reduction [i] based on digital (145, 259, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 72, 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 (73, 187, 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]

(256−111, 256, 280)-Net over F₄ — Digital

Digital (145, 256, 280)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−111, 256, 4354)-Net in Base 4 — Upper bound on s

There is no (145, 256, 4355)-net in base 4, because

1 times m-reduction [i] would yield (145, 255, 4355)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3378 552877 133651 670002 076520 554422 653883 273639 342976 356424 381334 904668 103782 536766 707335 890982 119122 351547 225511 862611 752994 942896 241957 284222 212511 819264 > 4²⁵⁵ [i]