MinT - Best Known (7, 7+5, s)-Nets in Base 5

Best Known (7, 7+5, s)-Nets in Base 5

(7, 7+5, 95)-Net over F₅ — Constructive and digital

Digital (7, 12, 95)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 31)-net over F₅, using
  - s-reduction based on digital (0, 1, s)-net over F₅ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (1, 3, 31)-net over F₅, using
  - kÂ =Â 2 construction [i]
3. digital (3, 8, 33)-net over F₅, using
  - linear code embedding found in a computer search [i]

(7, 7+5, 130)-Net over F₅ — Digital

Digital (7, 12, 130)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹², 130, F₅, 5) (dual of [130, 118, 6]-code), using
- a â€œGraâ€ code from Grasslâ€™s database [i]

(7, 7+5, 2469)-Net in Base 5 — Upper bound on s

There is no (7, 12, 2470)-net in base 5, because

1 times m-reduction [i] would yield (7, 11, 2470)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 48 846721 > 5¹¹ [i]