MinT - Best Known (48−7, 48, s)-Nets in Base 25

Best Known (48−7, 48, s)-Nets in Base 25

(48−7, 48, large)-Net over F₂₅ — Constructive and digital

Digital (41, 48, large)-net over F₂₅, using

25¹ times duplication [i] based on 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]

(48−7, 48, large)-Net in Base 25 — Upper bound on s

There is no (41, 48, large)-net in base 25, because

5 times m-reduction [i] would yield (41, 43, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (48−7, 48, s)-Nets in Base 25

(48−7, 48, large)-Net over F25 — Constructive and digital

(48−7, 48, large)-Net in Base 25 — Upper bound on s

(48−7, 48, large)-Net over F₂₅ — Constructive and digital