MinT - Best Known (89−6, 89, s)-Nets in Base 25

Best Known (89−6, 89, s)-Nets in Base 25

(89−6, 89, large)-Net over F₂₅ — Constructive and digital

Digital (83, 89, large)-net over F₂₅, using

5 times m-reduction [i] based on digital (83, 94, large)-net over F₂₅, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 1, 1677720)-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]
  2. digital (4, 6, 1677720)-net over F₂₅, using
    - s-reduction based on digital (4, 6, large)-net over F₂₅, using
      - kÂ =Â 2 construction [i]
  3. digital (4, 6, 1677720)-net over F₂₅ (see above)
  4. digital (6, 9, 1677720)-net over F₂₅, using
    - s-reduction based on digital (6, 9, large)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25⁹, large, F₂₅, 3, 3), using
        appending kth column [i] based on linear OOA(25⁹, large, F₂₅, 2, 3), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁹, large, F₂₅, 3) (dual of [large, large−9, 4]-code), using
        product of projective cap with (hyper)oval [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]
  5. 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]
  6. 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]

(89−6, 89, large)-Net in Base 25 — Upper bound on s

There is no (83, 89, large)-net in base 25, because

4 times m-reduction [i] would yield (83, 85, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (89−6, 89, s)-Nets in Base 25

(89−6, 89, large)-Net over F25 — Constructive and digital

(89−6, 89, large)-Net in Base 25 — Upper bound on s

(89−6, 89, large)-Net over F₂₅ — Constructive and digital