MinT - Best Known (49−11, 49, s)-Nets in Base 3

Best Known (49−11, 49, s)-Nets in Base 3

(49−11, 49, 400)-Net over F₃ — Constructive and digital

Digital (38, 49, 400)-net over F₃, using

3¹ times duplication [i] based on digital (37, 48, 400)-net over F₃, using
- trace code for nets [i] based on digital (1, 12, 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]

(49−11, 49, 719)-Net over F₃ — Digital

Digital (38, 49, 719)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁴⁹, 719, F₃, 11) (dual of [719, 670, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁴⁹, 1093, F₃, 11) (dual of [1093, 1044, 12]-code), using
  - the BCH-code C(I) with length 1093Â | 3⁷âˆ’1, defining interval IÂ =Â {1,3,5,…,21}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(49−11, 49, 49559)-Net in Base 3 — Upper bound on s

There is no (38, 49, 49560)-net in base 3, because

1 times m-reduction [i] would yield (38, 48, 49560)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 79766 837117 042577 348785 > 3⁴⁸ [i]