MinT - Best Known (84−3, 84, s)-Nets in Base 25

Best Known (84−3, 84, s)-Nets in Base 25

(84−3, 84, large)-Net over F₂₅ — Constructive and digital

Digital (81, 84, large)-net over F₂₅, using

t-expansion [i] based on digital (79, 84, large)-net over F₂₅, using
- 5 times m-reduction [i] based on digital (79, 89, 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, 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]
    17. digital (0, 1, 391875)-net over F₂₅ (see above)
    18. digital (0, 1, 391875)-net over F₂₅ (see above)
    19. digital (0, 1, 391875)-net over F₂₅ (see above)
    20. digital (0, 1, 391875)-net over F₂₅ (see above)
    21. 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]
    22. digital (3, 5, 391875)-net over F₂₅ (see above)
    23. 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]
    24. digital (16, 21, 1677720)-net over F₂₅, using
      - s-reduction based on digital (16, 21, 4194301)-net over F₂₅, using
        net defined by OOA [i] based on linear OOA(25²¹, 4194301, F₂₅, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(25²¹, large, F₂₅, 5) (dual of [large, large−21, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
    25. 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]

(84−3, 84, large)-Net in Base 25 — Upper bound on s

There is no (81, 84, large)-net in base 25, because

1 times m-reduction [i] would yield (81, 83, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (84−3, 84, s)-Nets in Base 25

(84−3, 84, large)-Net over F25 — Constructive and digital

(84−3, 84, large)-Net in Base 25 — Upper bound on s

(84−3, 84, large)-Net over F₂₅ — Constructive and digital