MinT - Best Known (258−114, 258, s)-Nets in Base 4

Best Known (258−114, 258, s)-Nets in Base 4

(258−114, 258, 134)-Net over F₄ — Constructive and digital

Digital (144, 258, 134)-net over F₄, using

2 times m-reduction [i] based on digital (144, 260, 134)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 71, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 189, 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]

(258−114, 258, 266)-Net over F₄ — Digital

Digital (144, 258, 266)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−114, 258, 3861)-Net in Base 4 — Upper bound on s

There is no (144, 258, 3862)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 216048 625492 082254 128864 373163 033172 373980 159158 172466 412257 502273 034270 993170 818755 525179 414413 745164 787594 667171 525537 964125 286010 983267 964968 344581 650720 > 4²⁵⁸ [i]