MinT - Best Known (89, 89+14, s)-Nets in Base 25

Best Known (89, 89+14, s)-Nets in Base 25

(89, 89+14, 2397042)-Net over F₂₅ — Constructive and digital

Digital (89, 103, 2397042)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 6, 300)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶, 300, F₂₅, 4, 4) (dual of [(300, 4), 1194, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(25⁶, 600, F₂₅, 4) (dual of [600, 594, 5]-code), using
      - 1 times truncation [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (24, 31, 1198371)-net over F₂₅, using
  - s-reduction based on digital (24, 31, 2796200)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25³¹, 2796200, F₂₅, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(25³¹, 8388601, F₂₅, 7) (dual of [8388601, 8388570, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(25³¹, large, F₂₅, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
3. digital (52, 66, 1198371)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁶⁶, 1198371, F₂₅, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(25⁶⁶, 8388597, F₂₅, 14) (dual of [8388597, 8388531, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(25⁶⁶, large, F₂₅, 14) (dual of [large, large−66, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(89, 89+14, large)-Net over F₂₅ — Digital

Digital (89, 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]

(89, 89+14, large)-Net in Base 25 — Upper bound on s

There is no (89, 103, large)-net in base 25, because

12 times m-reduction [i] would yield (89, 91, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (89, 89+14, s)-Nets in Base 25

(89, 89+14, 2397042)-Net over F25 — Constructive and digital

(89, 89+14, large)-Net over F25 — Digital

(89, 89+14, large)-Net in Base 25 — Upper bound on s

(89, 89+14, 2397042)-Net over F₂₅ — Constructive and digital

(89, 89+14, large)-Net over F₂₅ — Digital