MinT - Best Known (145, 145+113, s)-Nets in Base 4

Best Known (145, 145+113, s)-Nets in Base 4

(145, 145+113, 137)-Net over F₄ — Constructive and digital

Digital (145, 258, 137)-net over F₄, using

1 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]

(145, 145+113, 274)-Net over F₄ — Digital

Digital (145, 258, 274)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(145, 145+113, 4147)-Net in Base 4 — Upper bound on s

There is no (145, 258, 4148)-net in base 4, because

1 times m-reduction [i] would yield (145, 257, 4148)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54195 046788 795233 169068 552911 547873 544640 267245 169027 664306 954406 123785 697462 019206 774348 454643 598163 825111 925214 139813 836208 369668 359330 167168 082578 240000 > 4²⁵⁷ [i]