MinT - Best Known (41−3, 41, s)-Nets in Base 25

Best Known (41−3, 41, s)-Nets in Base 25

(41−3, 41, large)-Net over F₂₅ — Constructive and digital

Digital (38, 41, large)-net over F₂₅, using

3 times m-reduction [i] based on digital (38, 44, 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 (29, 35, 5592402)-net over F₂₅, using
    - (u,Â u+v)-construction [i] based on
      1. digital (6, 9, large)-net over F₂₅ (see above)
      2. 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]

(41−3, 41, large)-Net in Base 25 — Upper bound on s

There is no (38, 41, large)-net in base 25, because

1 times m-reduction [i] would yield (38, 40, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]