MinT - Best Known (248−95, 248, s)-Nets in Base 4

Best Known (248−95, 248, s)-Nets in Base 4

(248−95, 248, 160)-Net over F₄ — Constructive and digital

Digital (153, 248, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 80, 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, 168, 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]

(248−95, 248, 408)-Net over F₄ — Digital

Digital (153, 248, 408)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−95, 248, 8894)-Net in Base 4 — Upper bound on s

There is no (153, 248, 8895)-net in base 4, because

1 times m-reduction [i] would yield (153, 247, 8895)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51319 607490 472951 678842 215835 304387 120278 017677 640099 327958 649571 792849 708714 940676 502569 897149 141461 531141 915828 755517 170883 466053 358865 433737 695436 > 4²⁴⁷ [i]