MinT - Best Known (31−11, 31, s)-Nets in Base 4

Best Known (31−11, 31, s)-Nets in Base 4

(31−11, 31, 90)-Net over F₄ — Constructive and digital

Digital (20, 31, 90)-net over F₄, using

1 times m-reduction [i] based on digital (20, 32, 90)-net over F₄, using
- trace code for nets [i] based on digital (4, 16, 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]

(31−11, 31, 129)-Net over F₄ — Digital

Digital (20, 31, 129)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³¹, 129, F₄, 11) (dual of [129, 98, 12]-code), using
- a code from Grasslâ€™s database [i]

(31−11, 31, 3553)-Net in Base 4 — Upper bound on s

There is no (20, 31, 3554)-net in base 4, because

1 times m-reduction [i] would yield (20, 30, 3554)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 154118 406317 380824 > 4³⁰ [i]