MinT - Best Known (19, 40, s)-Nets in Base 49

Best Known (19, 40, s)-Nets in Base 49

(19, 40, 152)-Net over F₄₉ — Constructive and digital

Digital (19, 40, 152)-net over F₄₉, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 7, 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 (1, 11, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
3. digital (1, 22, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(19, 40, 479)-Net over F₄₉ — Digital

Digital (19, 40, 479)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁴⁰, 479, F₄₉, 21) (dual of [479, 439, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(49⁴⁰, 800, F₄₉, 21) (dual of [800, 760, 22]-code), using
  - the BCH-code C(I) with length 800Â | 49²âˆ’1, defining interval IÂ =Â {1,4,7,…,61}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(19, 40, 368548)-Net in Base 49 — Upper bound on s

There is no (19, 40, 368549)-net in base 49, because

1 times m-reduction [i] would yield (19, 39, 368549)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 827288 894417 077391 598169 210372 969560 658231 795620 682608 892631 138401 > 49³⁹ [i]