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

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

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

Digital (84, 87, large)-net over F₂₅, using

8 times m-reduction [i] based on digital (84, 95, 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, 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]
  16. digital (0, 1, 391875)-net over F₂₅ (see above)
  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 (40, 51, 1677720)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁵¹, 1677720, F₂₅, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(25⁵¹, 8388601, F₂₅, 11) (dual of [8388601, 8388550, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁵¹, large, F₂₅, 11) (dual of [large, large−51, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

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

There is no (84, 87, large)-net in base 25, because

1 times m-reduction [i] would yield (84, 86, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

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

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

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

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