MinT - Best Known (125, 125+83, s)-Nets in Base 4

Best Known (125, 125+83, s)-Nets in Base 4

(125, 125+83, 131)-Net over F₄ — Constructive and digital

Digital (125, 208, 131)-net over F₄, using

1 times m-reduction [i] based on digital (125, 209, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 52, 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, 157, 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]

(125, 125+83, 305)-Net over F₄ — Digital

Digital (125, 208, 305)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+83, 5861)-Net in Base 4 — Upper bound on s

There is no (125, 208, 5862)-net in base 4, because

1 times m-reduction [i] would yield (125, 207, 5862)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42576 544111 810045 386174 476468 014681 594933 854563 736803 180589 489185 892288 708069 453377 645797 276849 648581 795198 160621 168421 503000 > 4²⁰⁷ [i]