MinT - Best Known (36−13, 36, s)-Nets in Base 4

Best Known (36−13, 36, s)-Nets in Base 4

(36−13, 36, 98)-Net over F₄ — Constructive and digital

Digital (23, 36, 98)-net over F₄, using

trace code for nets [i] based on digital (5, 18, 49)-net over F₁₆, using
- net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
    - a function field by SÃ©mirat [i]

(36−13, 36, 128)-Net over F₄ — Digital

Digital (23, 36, 128)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁶, 128, F₄, 13) (dual of [128, 92, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4³⁶, 160, F₄, 13) (dual of [160, 124, 14]-code), using
  - a code from Grasslâ€™s database [i]

(36−13, 36, 3239)-Net in Base 4 — Upper bound on s

There is no (23, 36, 3240)-net in base 4, because

1 times m-reduction [i] would yield (23, 35, 3240)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1181 081026 258601 518495 > 4³⁵ [i]