MinT - Best Known (28, 35, s)-Nets in Base 25

Best Known (28, 35, s)-Nets in Base 25

(28, 35, 2796826)-Net over F₂₅ — Constructive and digital

Digital (28, 35, 2796826)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 4, 626)-net over F₂₅, using
  - net defined by OOA [i] based on linear OOA(25⁴, 626, F₂₅, 3, 3) (dual of [(626, 3), 1874, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(25⁴, 626, F₂₅, 2, 3) (dual of [(626, 2), 1248, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁴, 626, F₂₅, 3) (dual of [626, 622, 4]-code or 626-cap in PG(3,25)), using
        ovoid in PG(3,Â 25) [i]
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]

(28, 35, large)-Net over F₂₅ — Digital

Digital (28, 35, large)-net over F₂₅, using

1 times m-reduction [i] based on digital (28, 36, large)-net over F₂₅, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25³⁶, large, F₂₅, 8) (dual of [large, large−36, 9]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(28, 35, large)-Net in Base 25 — Upper bound on s

There is no (28, 35, large)-net in base 25, because

5 times m-reduction [i] would yield (28, 30, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]