MinT - Best Known (1, 1+3, s)-Nets in Base 25

Best Known (1, 1+3, s)-Nets in Base 25

(1, 1+3, 626)-Net over F₂₅ — Constructive and digital

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]

(1, 1+3, 626)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (1, 4, 627)-net over F₂₅, because

extracting embedded orthogonal array [i] would yield linear OA(25⁴, 627, F₂₅, 3) (dual of [627, 623, 4]-code or 627-cap in PG(3,25)), but
- ovoid cap is optimal [i]

(1, 1+3, 650)-Net in Base 25 — Upper bound on s

There is no (1, 4, 651)-net in base 25, because

extracting embedded orthogonal array [i] would yield OA(25⁴, 651, S₂₅, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]