MinT - Best Known (157, 157+99, s)-Nets in Base 4

Best Known (157, 157+99, s)-Nets in Base 4

(157, 157+99, 160)-Net over F₄ — Constructive and digital

Digital (157, 256, 160)-net over F₄, using

3 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 84, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 175, 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]

(157, 157+99, 407)-Net over F₄ — Digital

Digital (157, 256, 407)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+99, 8616)-Net in Base 4 — Upper bound on s

There is no (157, 256, 8617)-net in base 4, because

1 times m-reduction [i] would yield (157, 255, 8617)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3361 112871 562061 466692 900727 393677 655531 711414 157414 131997 257436 600728 729462 032180 262895 178451 322673 525201 250756 035982 639199 762038 404490 622380 225951 566176 > 4²⁵⁵ [i]