MinT - Best Known (61−10, 61, s)-Nets in Base 25

Best Known (61−10, 61, s)-Nets in Base 25

(61−10, 61, 1685870)-Net over F₂₅ — Constructive and digital

Digital (51, 61, 1685870)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (10, 15, 8150)-net over F₂₅, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 0, 326)-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, 326)-net over F₂₅ (see above)
    3. digital (0, 0, 326)-net over F₂₅ (see above)
    4. digital (0, 0, 326)-net over F₂₅ (see above)
    5. digital (0, 0, 326)-net over F₂₅ (see above)
    6. digital (0, 0, 326)-net over F₂₅ (see above)
    7. digital (0, 0, 326)-net over F₂₅ (see above)
    8. digital (0, 0, 326)-net over F₂₅ (see above)
    9. digital (0, 0, 326)-net over F₂₅ (see above)
    10. digital (0, 0, 326)-net over F₂₅ (see above)
    11. digital (0, 0, 326)-net over F₂₅ (see above)
    12. digital (0, 0, 326)-net over F₂₅ (see above)
    13. digital (0, 0, 326)-net over F₂₅ (see above)
    14. digital (0, 0, 326)-net over F₂₅ (see above)
    15. digital (0, 0, 326)-net over F₂₅ (see above)
    16. digital (0, 0, 326)-net over F₂₅ (see above)
    17. digital (0, 0, 326)-net over F₂₅ (see above)
    18. digital (0, 0, 326)-net over F₂₅ (see above)
    19. digital (0, 0, 326)-net over F₂₅ (see above)
    20. digital (0, 0, 326)-net over F₂₅ (see above)
    21. digital (0, 1, 326)-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]
    22. digital (0, 1, 326)-net over F₂₅ (see above)
    23. digital (0, 1, 326)-net over F₂₅ (see above)
    24. digital (1, 3, 326)-net over F₂₅, using
      - s-reduction based on digital (1, 3, 651)-net over F₂₅, using
        kÂ =Â 2 construction [i]
    25. digital (4, 9, 326)-net over F₂₅, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 2, 26)-net over F₂₅, using
        kÂ =Â 2 construction [i]
        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 (36, 46, 1677720)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁴⁶, 1677720, F₂₅, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(25⁴⁶, 8388600, F₂₅, 10) (dual of [8388600, 8388554, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁴⁶, large, F₂₅, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(61−10, 61, large)-Net over F₂₅ — Digital

Digital (51, 61, large)-net over F₂₅, using

t-expansion [i] based on digital (48, 61, large)-net over F₂₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁶¹, large, F₂₅, 13) (dual of [large, large−61, 14]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(61−10, 61, large)-Net in Base 25 — Upper bound on s

There is no (51, 61, large)-net in base 25, because

8 times m-reduction [i] would yield (51, 53, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (61−10, 61, s)-Nets in Base 25

(61−10, 61, 1685870)-Net over F25 — Constructive and digital

(61−10, 61, large)-Net over F25 — Digital

(61−10, 61, large)-Net in Base 25 — Upper bound on s

(61−10, 61, 1685870)-Net over F₂₅ — Constructive and digital

(61−10, 61, large)-Net over F₂₅ — Digital