MinT - Best Known (130, 130+85, s)-Nets in Base 4

Best Known (130, 130+85, s)-Nets in Base 4

(130, 130+85, 137)-Net over F₄ — Constructive and digital

Digital (130, 215, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 57, 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, 158, 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]

(130, 130+85, 324)-Net over F₄ — Digital

Digital (130, 215, 324)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(130, 130+85, 6397)-Net in Base 4 — Upper bound on s

There is no (130, 215, 6398)-net in base 4, because

1 times m-reduction [i] would yield (130, 214, 6398)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 696 891824 388482 247156 799768 021832 384583 451426 552698 678368 584488 232153 696959 071703 755176 828983 856969 664472 649156 703097 954717 620122 > 4²¹⁴ [i]