MinT - Best Known (251−106, 251, s)-Nets in Base 4

Best Known (251−106, 251, s)-Nets in Base 4

(251−106, 251, 137)-Net over F₄ — Constructive and digital

Digital (145, 251, 137)-net over F₄, using

8 times m-reduction [i] based on digital (145, 259, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 72, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 187, 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]

(251−106, 251, 300)-Net over F₄ — Digital

Digital (145, 251, 300)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−106, 251, 4831)-Net in Base 4 — Upper bound on s

There is no (145, 251, 4832)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 231274 645942 020101 044935 266716 413375 079478 298042 911489 033534 266605 580208 931557 970189 143384 021202 516174 957216 026021 786063 535618 085669 816611 063228 149610 > 4²⁵¹ [i]