MinT - Best Known (248−106, 248, s)-Nets in Base 4

Best Known (248−106, 248, s)-Nets in Base 4

(248−106, 248, 137)-Net over F₄ — Constructive and digital

Digital (142, 248, 137)-net over F₄, using

2 times m-reduction [i] based on digital (142, 250, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 69, 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, 181, 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]

(248−106, 248, 286)-Net over F₄ — Digital

Digital (142, 248, 286)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−106, 248, 4463)-Net in Base 4 — Upper bound on s

There is no (142, 248, 4464)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 206597 584555 994770 015492 893341 935189 973200 221810 368149 247705 109082 949463 543893 463428 740775 972921 645676 947051 845182 294442 388283 901634 707329 100371 692500 > 4²⁴⁸ [i]