MinT - Best Known (43−12, 43, s)-Nets in Base 49

Best Known (43−12, 43, s)-Nets in Base 49

(43−12, 43, 19708)-Net over F₄₉ — Constructive and digital

Digital (31, 43, 19708)-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 (22, 34, 19608)-net over F₄₉, using
  - net defined by OOA [i] based on linear OOA(49³⁴, 19608, F₄₉, 12, 12) (dual of [(19608, 12), 235262, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(49³⁴, 117648, F₄₉, 12) (dual of [117648, 117614, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(49³⁴, 117649, F₄₉, 12) (dual of [117649, 117615, 13]-code), using
        an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 117648Â = 49³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(43−12, 43, 413913)-Net over F₄₉ — Digital

Digital (31, 43, 413913)-net over F₄₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(43−12, 43, large)-Net in Base 49 — Upper bound on s

There is no (31, 43, large)-net in base 49, because

10 times m-reduction [i] would yield (31, 33, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]