MinT - Best Known (68−7, 68, s)-Nets in Base 25

Best Known (68−7, 68, s)-Nets in Base 25

(68−7, 68, large)-Net over F₂₅ — Constructive and digital

Digital (61, 68, large)-net over F₂₅, using

25² times duplication [i] based on digital (59, 66, large)-net over F₂₅, using
- t-expansion [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]

(68−7, 68, large)-Net in Base 25 — Upper bound on s

There is no (61, 68, large)-net in base 25, because

5 times m-reduction [i] would yield (61, 63, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (68−7, 68, s)-Nets in Base 25

(68−7, 68, large)-Net over F25 — Constructive and digital

(68−7, 68, large)-Net in Base 25 — Upper bound on s

(68−7, 68, large)-Net over F₂₅ — Constructive and digital