MinT - Best Known (67−9, 67, s)-Nets in Base 25

Best Known (67−9, 67, s)-Nets in Base 25

(67−9, 67, 6291450)-Net over F₂₅ — Constructive and digital

Digital (58, 67, 6291450)-net over F₂₅, using

25¹ times duplication [i] based on digital (57, 66, 6291450)-net over F₂₅, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (6, 9, 2097150)-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]
  2. 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]
  3. digital (32, 41, 2097150)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁴¹, 2097150, F₂₅, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OA(25⁴¹, 8388601, F₂₅, 9) (dual of [8388601, 8388560, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁴¹, large, F₂₅, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(67−9, 67, large)-Net over F₂₅ — Digital

Digital (58, 67, large)-net over F₂₅, using

t-expansion [i] based on digital (57, 67, large)-net over F₂₅, using
- 5 times m-reduction [i] based on digital (57, 72, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁷², large, F₂₅, 15) (dual of [large, large−72, 16]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25⁷¹, large, F₂₅, 15) (dual of [large, large−71, 16]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]

(67−9, 67, large)-Net in Base 25 — Upper bound on s

There is no (58, 67, large)-net in base 25, because

7 times m-reduction [i] would yield (58, 60, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (67−9, 67, s)-Nets in Base 25

(67−9, 67, 6291450)-Net over F25 — Constructive and digital

(67−9, 67, large)-Net over F25 — Digital

(67−9, 67, large)-Net in Base 25 — Upper bound on s

(67−9, 67, 6291450)-Net over F₂₅ — Constructive and digital

(67−9, 67, large)-Net over F₂₅ — Digital