MinT - Best Known (259−117, 259, s)-Nets in Base 4

Best Known (259−117, 259, s)-Nets in Base 4

(259−117, 259, 131)-Net over F₄ — Constructive and digital

Digital (142, 259, 131)-net over F₄, using

1 times m-reduction [i] based on digital (142, 260, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 69, 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, 191, 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]

(259−117, 259, 249)-Net over F₄ — Digital

Digital (142, 259, 249)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−117, 259, 3518)-Net in Base 4 — Upper bound on s

There is no (142, 259, 3519)-net in base 4, because

1 times m-reduction [i] would yield (142, 258, 3519)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 215052 693878 597152 882325 062134 589260 424740 797420 987839 803211 493837 725926 091682 141986 784273 581154 117943 980804 496825 032064 651838 030478 942058 174241 772979 160441 > 4²⁵⁸ [i]