MinT - Best Known (18, 18+10, s)-Nets in Base 4

Best Known (18, 18+10, s)-Nets in Base 4

(18, 18+10, 90)-Net over F₄ — Constructive and digital

Digital (18, 28, 90)-net over F₄, using

trace code for nets [i] based on digital (4, 14, 45)-net over F₁₆, using
- net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
    - a function field by GarcÃa and Quoos [i]

(18, 18+10, 130)-Net over F₄ — Digital

Digital (18, 28, 130)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁸, 130, F₄, 10) (dual of [130, 102, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁸, 144, F₄, 10) (dual of [144, 116, 11]-code), using
  - a code from Grasslâ€™s database [i]

(18, 18+10, 2039)-Net in Base 4 — Upper bound on s

There is no (18, 28, 2040)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 72189 246012 254695 > 4²⁸ [i]