MinT - Best Known (50−11, 50, s)-Nets in Base 25

Best Known (50−11, 50, s)-Nets in Base 25

(50−11, 50, 78451)-Net over F₂₅ — Constructive and digital

Digital (39, 50, 78451)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 9, 326)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 26)-net over F₂₅, using
      - kÂ =Â 2 construction [i]
    2. digital (2, 7, 300)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25⁷, 300, F₂₅, 5, 5) (dual of [(300, 5), 1493, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (30, 41, 78125)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁴¹, 78125, F₂₅, 11, 11) (dual of [(78125, 11), 859334, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(25⁴¹, 390626, F₂₅, 11) (dual of [390626, 390585, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 390626Â | 25⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(50−11, 50, 1842749)-Net over F₂₅ — Digital

Digital (39, 50, 1842749)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(50−11, 50, large)-Net in Base 25 — Upper bound on s

There is no (39, 50, large)-net in base 25, because

9 times m-reduction [i] would yield (39, 41, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]