MinT - Best Known (56−13, 56, s)-Nets in Base 3

Best Known (56−13, 56, s)-Nets in Base 3

(56−13, 56, 400)-Net over F₃ — Constructive and digital

Digital (43, 56, 400)-net over F₃, using

trace code for nets [i] based on digital (1, 14, 100)-net over F₈₁, using
- net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
    - a function field by SÃ©mirat [i]

(56−13, 56, 587)-Net over F₃ — Digital

Digital (43, 56, 587)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵⁶, 587, F₃, 13) (dual of [587, 531, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵⁶, 1093, F₃, 13) (dual of [1093, 1037, 14]-code), using
  - the BCH-code C(I) with length 1093Â | 3⁷âˆ’1, defining interval IÂ =Â {1,105,209,…,156}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(56−13, 56, 35378)-Net in Base 3 — Upper bound on s

There is no (43, 56, 35379)-net in base 3, because

1 times m-reduction [i] would yield (43, 55, 35379)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 174 472036 474294 276467 652477 > 3⁵⁵ [i]