MinT - Best Known (45−3, 45, s)-Nets in Base 25

Best Known (45−3, 45, s)-Nets in Base 25

(45−3, 45, large)-Net over F₂₅ — Constructive and digital

Digital (42, 45, large)-net over F₂₅, using

4 times m-reduction [i] based on digital (42, 49, large)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. 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 (33, 40, 5592400)-net over F₂₅, using
    - (u,Â u+v)-construction [i] based on
      1. digital (6, 9, large)-net over F₂₅ (see above)
      2. 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]

(45−3, 45, large)-Net in Base 25 — Upper bound on s

There is no (42, 45, large)-net in base 25, because

1 times m-reduction [i] would yield (42, 44, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (45−3, 45, s)-Nets in Base 25

(45−3, 45, large)-Net over F25 — Constructive and digital

(45−3, 45, large)-Net in Base 25 — Upper bound on s

(45−3, 45, large)-Net over F₂₅ — Constructive and digital