MinT - Best Known (70−8, 70, s)-Nets in Base 25

Best Known (70−8, 70, s)-Nets in Base 25

(70−8, 70, large)-Net over F₂₅ — Constructive and digital

Digital (62, 70, large)-net over F₂₅, using

25⁴ times duplication [i] based on digital (58, 66, large)-net over F₂₅, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 406901)-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, 406901)-net over F₂₅ (see above)
  3. digital (0, 0, 406901)-net over F₂₅ (see above)
  4. digital (0, 0, 406901)-net over F₂₅ (see above)
  5. digital (0, 0, 406901)-net over F₂₅ (see above)
  6. digital (0, 0, 406901)-net over F₂₅ (see above)
  7. digital (0, 0, 406901)-net over F₂₅ (see above)
  8. digital (0, 0, 406901)-net over F₂₅ (see above)
  9. digital (0, 0, 406901)-net over F₂₅ (see above)
  10. digital (0, 0, 406901)-net over F₂₅ (see above)
  11. digital (0, 0, 406901)-net over F₂₅ (see above)
  12. digital (0, 0, 406901)-net over F₂₅ (see above)
  13. digital (0, 0, 406901)-net over F₂₅ (see above)
  14. digital (0, 0, 406901)-net over F₂₅ (see above)
  15. digital (0, 0, 406901)-net over F₂₅ (see above)
  16. digital (0, 0, 406901)-net over F₂₅ (see above)
  17. digital (0, 0, 406901)-net over F₂₅ (see above)
  18. digital (0, 1, 406901)-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]
  19. digital (0, 1, 406901)-net over F₂₅ (see above)
  20. digital (0, 1, 406901)-net over F₂₅ (see above)
  21. digital (0, 1, 406901)-net over F₂₅ (see above)
  22. digital (3, 5, 406901)-net over F₂₅, using
    - kÂ =Â 2 construction [i]
  23. digital (3, 5, 406901)-net over F₂₅ (see above)
  24. digital (12, 16, 2097150)-net over F₂₅, using
    - s-reduction based on digital (12, 16, 4194301)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25¹⁶, 4194301, F₂₅, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(25¹⁶, 8388602, F₂₅, 4) (dual of [8388602, 8388586, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹⁶, large, F₂₅, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
  25. digital (28, 36, 2097150)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25³⁶, 2097150, F₂₅, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
      - OA 4-folding and stacking [i] based on linear OA(25³⁶, 8388600, F₂₅, 8) (dual of [8388600, 8388564, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(25³⁶, large, F₂₅, 8) (dual of [large, large−36, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(70−8, 70, large)-Net in Base 25 — Upper bound on s

There is no (62, 70, large)-net in base 25, because

6 times m-reduction [i] would yield (62, 64, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (70−8, 70, s)-Nets in Base 25

(70−8, 70, large)-Net over F25 — Constructive and digital

(70−8, 70, large)-Net in Base 25 — Upper bound on s

(70−8, 70, large)-Net over F₂₅ — Constructive and digital