MinT - Best Known (251−97, 251, s)-Nets in Base 4

Best Known (251−97, 251, s)-Nets in Base 4

(251−97, 251, 160)-Net over F₄ — Constructive and digital

Digital (154, 251, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 81, 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, 170, 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]

(251−97, 251, 401)-Net over F₄ — Digital

Digital (154, 251, 401)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−97, 251, 8499)-Net in Base 4 — Upper bound on s

There is no (154, 251, 8500)-net in base 4, because

1 times m-reduction [i] would yield (154, 250, 8500)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 284720 162493 895777 004845 505703 206671 746057 957880 845929 492174 467805 875523 719508 802593 283206 937128 944452 308019 737799 779232 910378 769474 503077 805093 648321 > 4²⁵⁰ [i]