MinT - Best Known (53−27, 53, s)-Nets in Base 25

Best Known (53−27, 53, s)-Nets in Base 25

(53−27, 53, 200)-Net over F₂₅ — Constructive and digital

Digital (26, 53, 200)-net over F₂₅, using

t-expansion [i] based on digital (25, 53, 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]

(53−27, 53, 332)-Net over F₂₅ — Digital

Digital (26, 53, 332)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁵³, 332, F₂₅, 27) (dual of [332, 279, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁵³, 624, F₂₅, 27) (dual of [624, 571, 28]-code), using
  - the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {3,4,…,29}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

(53−27, 53, 92241)-Net in Base 25 — Upper bound on s

There is no (26, 53, 92242)-net in base 25, because

1 times m-reduction [i] would yield (26, 52, 92242)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 4 931050 372615 172193 711390 810460 061892 826869 020146 123581 620715 087072 906865 > 25⁵² [i]