MinT - Best Known (47−7, 47, s)-Nets in Base 25

Best Known (47−7, 47, s)-Nets in Base 25

(47−7, 47, large)-Net over F₂₅ — Constructive and digital

Digital (40, 47, large)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 0, 391875)-net over F₂₅, using
  - s-reduction based on digital (0, 0, s)-net over F₂₅ with arbitrarily large s, using
    - arbitrary set of 25⁰ points [i]
2. digital (0, 0, 391875)-net over F₂₅ (see above)
3. digital (0, 0, 391875)-net over F₂₅ (see above)
4. digital (0, 0, 391875)-net over F₂₅ (see above)
5. digital (0, 0, 391875)-net over F₂₅ (see above)
6. digital (0, 0, 391875)-net over F₂₅ (see above)
7. digital (0, 0, 391875)-net over F₂₅ (see above)
8. digital (0, 0, 391875)-net over F₂₅ (see above)
9. digital (0, 0, 391875)-net over F₂₅ (see above)
10. digital (0, 0, 391875)-net over F₂₅ (see above)
11. digital (0, 0, 391875)-net over F₂₅ (see above)
12. digital (0, 0, 391875)-net over F₂₅ (see above)
13. digital (0, 0, 391875)-net over F₂₅ (see above)
14. digital (0, 0, 391875)-net over F₂₅ (see above)
15. digital (0, 0, 391875)-net over F₂₅ (see above)
16. digital (0, 0, 391875)-net over F₂₅ (see above)
17. digital (0, 0, 391875)-net over F₂₅ (see above)
18. digital (0, 0, 391875)-net over F₂₅ (see above)
19. digital (0, 1, 391875)-net over F₂₅, using
  - s-reduction based on digital (0, 1, s)-net over F₂₅ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
20. digital (0, 1, 391875)-net over F₂₅ (see above)
21. digital (0, 1, 391875)-net over F₂₅ (see above)
22. digital (0, 1, 391875)-net over F₂₅ (see above)
23. digital (3, 5, 391875)-net over F₂₅, using
  - s-reduction based on digital (3, 5, 406901)-net over F₂₅, using
    - kÂ =Â 2 construction [i]
24. digital (4, 7, 391875)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁷, 391875, F₂₅, 3, 3) (dual of [(391875, 3), 1175618, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(25⁷, 391875, F₂₅, 2, 3) (dual of [(391875, 2), 783743, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁷, 391875, F₂₅, 3) (dual of [391875, 391868, 4]-code or 391875-cap in PG(6,25)), using
        product of two caps with tangent hyperplane [i]
25. digital (24, 31, 2796200)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25³¹, 2796200, F₂₅, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(25³¹, 8388601, F₂₅, 7) (dual of [8388601, 8388570, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(25³¹, large, F₂₅, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(47−7, 47, large)-Net in Base 25 — Upper bound on s

There is no (40, 47, large)-net in base 25, because

5 times m-reduction [i] would yield (40, 42, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (47−7, 47, s)-Nets in Base 25

(47−7, 47, large)-Net over F25 — Constructive and digital

(47−7, 47, large)-Net in Base 25 — Upper bound on s

(47−7, 47, large)-Net over F₂₅ — Constructive and digital