MinT - Best Known (9, 23, s)-Nets in Base 49

Best Known (9, 23, s)-Nets in Base 49

(9, 23, 102)-Net over F₄₉ — Constructive and digital

Digital (9, 23, 102)-net over F₄₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]
2. digital (1, 15, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(9, 23, 134)-Net over F₄₉ — Digital

Digital (9, 23, 134)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49²³, 134, F₄₉, 14) (dual of [134, 111, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(49²³, 150, F₄₉, 14) (dual of [150, 127, 15]-code), using
  - the BCH-code C(I) with length 150Â | 49²âˆ’1, defining interval IÂ =Â {1,8,15,…,92}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(9, 23, 25183)-Net in Base 49 — Upper bound on s

There is no (9, 23, 25184)-net in base 49, because

the generalized Rao bound for nets shows that 49^mÂ â‰¥ 749 050003 912080 696762 426107 240091 323905 > 49²³ [i]