MinT - Best Known (103−13, 103, s)-Nets in Base 25

Best Known (103−13, 103, s)-Nets in Base 25

(103−13, 103, 4194300)-Net over F₂₅ — Constructive and digital

Digital (90, 103, 4194300)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (12, 16, 1398100)-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]
2. digital (20, 26, 1398100)-net over F₂₅, using
  - s-reduction based on digital (20, 26, 2796201)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25²⁶, 2796201, F₂₅, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(25²⁶, large, F₂₅, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. digital (48, 61, 1398100)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶¹, 1398100, F₂₅, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(25⁶¹, 8388601, F₂₅, 13) (dual of [8388601, 8388540, 14]-code), using
      - discarding factors / shortening the dual code 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]

(103−13, 103, large)-Net over F₂₅ — Digital

Digital (90, 103, large)-net over F₂₅, using

t-expansion [i] based on digital (85, 103, large)-net over F₂₅, using
- 4 times m-reduction [i] based on digital (85, 107, large)-net over F₂₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25¹⁰⁷, large, F₂₅, 22) (dual of [large, large−107, 23]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(25¹⁰⁶, large, F₂₅, 22) (dual of [large, large−106, 23]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(103−13, 103, large)-Net in Base 25 — Upper bound on s

There is no (90, 103, large)-net in base 25, because

11 times m-reduction [i] would yield (90, 92, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (103−13, 103, s)-Nets in Base 25

(103−13, 103, 4194300)-Net over F25 — Constructive and digital

(103−13, 103, large)-Net over F25 — Digital

(103−13, 103, large)-Net in Base 25 — Upper bound on s

(103−13, 103, 4194300)-Net over F₂₅ — Constructive and digital

(103−13, 103, large)-Net over F₂₅ — Digital