MinT - Best Known (54−12, 54, s)-Nets in Base 49

Best Known (54−12, 54, s)-Nets in Base 49

(54−12, 54, 960900)-Net over F₄₉ — Constructive and digital

Digital (42, 54, 960900)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 100)-net over F₄₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 50)-net over F₄₉, using
      - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 6, 50)-net over F₄₉, using
      - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)
2. digital (33, 45, 960800)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49⁴⁵, 960800, F₄₉, 12, 12) (dual of [(960800, 12), 11529555, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(49⁴⁵, 5764800, F₄₉, 12) (dual of [5764800, 5764755, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(49⁴⁵, 5764801, F₄₉, 12) (dual of [5764801, 5764756, 13]-code), using
        an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 5764800Â = 49⁴âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(54−12, 54, large)-Net over F₄₉ — Digital

Digital (42, 54, large)-net over F₄₉, using

49² times duplication [i] based on digital (40, 52, large)-net over F₄₉, using
- Gilbertâ€“VarÅ¡amov bound [i]

(54−12, 54, large)-Net in Base 49 — Upper bound on s

There is no (42, 54, large)-net in base 49, because

10 times m-reduction [i] would yield (42, 44, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]