MinT - Best Known (20, 20+23, s)-Nets in Base 49

Best Known (20, 20+23, s)-Nets in Base 49

(20, 20+23, 152)-Net over F₄₉ — Constructive and digital

Digital (20, 43, 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, 12, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
3. digital (1, 24, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(20, 20+23, 425)-Net over F₄₉ — Digital

Digital (20, 43, 425)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49⁴³, 425, F₄₉, 23) (dual of [425, 382, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(49⁴³, 480, F₄₉, 23) (dual of [480, 437, 24]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 480Â | 49²âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(20, 20+23, 290563)-Net in Base 49 — Upper bound on s

There is no (20, 43, 290564)-net in base 49, because

1 times m-reduction [i] would yield (20, 42, 290564)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 97329 328446 682466 453308 923875 227872 202969 010207 805121 730207 278675 234369 > 49⁴² [i]