MinT - Best Known (257−99, 257, s)-Nets in Base 4

Best Known (257−99, 257, s)-Nets in Base 4

(257−99, 257, 160)-Net over F₄ — Constructive and digital

Digital (158, 257, 160)-net over F₄, using

t-expansion [i] based on digital (157, 257, 160)-net over F₄, using
- 2 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]

(257−99, 257, 414)-Net over F₄ — Digital

Digital (158, 257, 414)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−99, 257, 8864)-Net in Base 4 — Upper bound on s

There is no (158, 257, 8865)-net in base 4, because

1 times m-reduction [i] would yield (158, 256, 8865)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13413 087567 491849 722440 451474 518697 941710 434315 427647 634656 163803 948834 132965 042421 137564 194721 485940 954475 523662 088393 816813 922138 975586 734805 762028 238496 > 4²⁵⁶ [i]