MinT - Best Known (12, 29, s)-Nets in Base 49

Best Known (12, 29, s)-Nets in Base 49

(12, 29, 104)-Net over F₄₉ — Constructive and digital

Digital (12, 29, 104)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (2, 19, 52)-net over F₄₉, using
  - net from sequence [i] based on digital (2, 51)-sequence over F₄₉ (see above)

(12, 29, 185)-Net over F₄₉ — Digital

Digital (12, 29, 185)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49²⁹, 185, F₄₉, 17) (dual of [185, 156, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(49²⁹, 200, F₄₉, 17) (dual of [200, 171, 18]-code), using
  - the BCH-code C(I) with length 200Â | 49²âˆ’1, defining interval IÂ =Â {1,4,7,…,49}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(12, 29, 64581)-Net in Base 49 — Upper bound on s

There is no (12, 29, 64582)-net in base 49, because

1 times m-reduction [i] would yield (12, 28, 64582)-net in base 49, but
- the generalized Rao bound for nets shows that 49^mÂ â‰¥ 211591 660805 635874 411809 355765 093878 103614 199041 > 49²⁸ [i]