MinT - Best Known (125, 125+81, s)-Nets in Base 4

Best Known (125, 125+81, s)-Nets in Base 4

(125, 125+81, 132)-Net over F₄ — Constructive and digital

Digital (125, 206, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 52, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 154, 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]

(125, 125+81, 317)-Net over F₄ — Digital

Digital (125, 206, 317)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+81, 6367)-Net in Base 4 — Upper bound on s

There is no (125, 206, 6368)-net in base 4, because

1 times m-reduction [i] would yield (125, 205, 6368)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2645 398635 004685 184916 741998 180086 929889 239809 949574 300045 868183 188605 026845 498114 562824 502130 705389 243336 671237 010787 215382 > 4²⁰⁵ [i]