MinT - Best Known (245−91, 245, s)-Nets in Base 4

Best Known (245−91, 245, s)-Nets in Base 4

(245−91, 245, 160)-Net over F₄ — Constructive and digital

Digital (154, 245, 160)-net over F₄, using

5 times m-reduction [i] based on digital (154, 250, 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, 169, 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]

(245−91, 245, 449)-Net over F₄ — Digital

Digital (154, 245, 449)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−91, 245, 10766)-Net in Base 4 — Upper bound on s

There is no (154, 245, 10767)-net in base 4, because

1 times m-reduction [i] would yield (154, 244, 10767)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 801 424976 666134 648095 225794 763774 320037 544569 666611 239133 940110 094400 884581 647174 526669 667543 285361 245484 936720 125278 053656 909641 582155 195575 887776 > 4²⁴⁴ [i]