MinT - Best Known (246−109, 246, s)-Nets in Base 4

Best Known (246−109, 246, s)-Nets in Base 4

(246−109, 246, 131)-Net over F₄ — Constructive and digital

Digital (137, 246, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 64, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 182, 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]

(246−109, 246, 253)-Net over F₄ — Digital

Digital (137, 246, 253)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−109, 246, 3722)-Net in Base 4 — Upper bound on s

There is no (137, 246, 3723)-net in base 4, because

1 times m-reduction [i] would yield (137, 245, 3723)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3212 640993 371205 310924 817115 860615 964354 411817 714682 671284 555782 555602 861054 743617 830297 376056 529193 980337 966203 920979 158723 994918 944196 452911 321040 > 4²⁴⁵ [i]