MinT - Best Known (15−9, 15, s)-Nets in Base 25

Best Known (15−9, 15, s)-Nets in Base 25

(15−9, 15, 66)-Net over F₂₅ — Constructive and digital

Digital (6, 15, 66)-net over F₂₅, using

t-expansion [i] based on digital (4, 15, 66)-net over F₂₅, using
- net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
    - a function field by GarcÃa and Quoos [i]

(15−9, 15, 86)-Net over F₂₅ — Digital

Digital (6, 15, 86)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25¹⁵, 86, F₂₅, 9) (dual of [86, 71, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(25¹⁵, 104, F₂₅, 9) (dual of [104, 89, 10]-code), using
  - the BCH-code C(I) with length 104Â | 25²âˆ’1, defining interval IÂ =Â {1,4,7,…,25}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(15−9, 15, 7203)-Net in Base 25 — Upper bound on s

There is no (6, 15, 7204)-net in base 25, because

1 times m-reduction [i] would yield (6, 14, 7204)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 37 267509 158063 389825 > 25¹⁴ [i]