MinT - Best Known (25, 51, s)-Nets in Base 25

Best Known (25, 51, s)-Nets in Base 25

(25, 51, 200)-Net over F₂₅ — Constructive and digital

Digital (25, 51, 200)-net over F₂₅, using

net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
  - a function field by GarcÃa and Quoos [i]

(25, 51, 323)-Net over F₂₅ — Digital

Digital (25, 51, 323)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁵¹, 323, F₂₅, 26) (dual of [323, 272, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁵¹, 624, F₂₅, 26) (dual of [624, 573, 27]-code), using
  - the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {3,4,…,28}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(25, 51, 72008)-Net in Base 25 — Upper bound on s

There is no (25, 51, 72009)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 197245 373451 157738 864027 153427 135027 126329 360175 680918 265338 795628 834425 > 25⁵¹ [i]