MinT - Best Known (40−11, 40, s)-Nets in Base 25

Best Known (40−11, 40, s)-Nets in Base 25

(40−11, 40, 3451)-Net over F₂₅ — Constructive and digital

Digital (29, 40, 3451)-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 (20, 31, 3125)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25³¹, 3125, F₂₅, 11, 11) (dual of [(3125, 11), 34344, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(25³¹, 15626, F₂₅, 11) (dual of [15626, 15595, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 15626Â | 25⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(40−11, 40, 73715)-Net over F₂₅ — Digital

Digital (29, 40, 73715)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(40−11, 40, large)-Net in Base 25 — Upper bound on s

There is no (29, 40, large)-net in base 25, because

9 times m-reduction [i] would yield (29, 31, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]