MinT - Best Known (5−3, 5, s)-Nets in Base 5

Best Known (5−3, 5, s)-Nets in Base 5

(5−3, 5, 66)-Net over F₅ — Constructive and digital

Digital (2, 5, 66)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁵, 66, F₅, 3, 3) (dual of [(66, 3), 193, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(5⁵, 66, F₅, 2, 3) (dual of [(66, 2), 127, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁵, 66, F₅, 3) (dual of [66, 61, 4]-code or 66-cap in PG(4,5)), using
    - caps completed using a computer search [i]

(5−3, 5, 88)-Net over F₅ — Upper bound on s (digital)

There is no digital (2, 5, 89)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5⁵, 89, F₅, 3) (dual of [89, 84, 4]-code or 89-cap in PG(4,5)), but
- a result by Edel, Storme, and Sziklai [i]

(5−3, 5, 155)-Net in Base 5 — Upper bound on s

There is no (2, 5, 156)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁵, 156, S₅, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]