MinT - Best Known (3, 6, s)-Nets in Base 5

Best Known (3, 6, s)-Nets in Base 5

(3, 6, 186)-Net over F₅ — Constructive and digital

Digital (3, 6, 186)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁶, 186, F₅, 3, 3) (dual of [(186, 3), 552, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(5⁶, 186, F₅, 2, 3) (dual of [(186, 2), 366, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁶, 186, F₅, 3) (dual of [186, 180, 4]-code or 186-cap in PG(5,5)), using
    - caps completed using a computer search [i]

(3, 6, 434)-Net over F₅ — Upper bound on s (digital)

There is no digital (3, 6, 435)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5⁶, 435, F₅, 3) (dual of [435, 429, 4]-code or 435-cap in PG(5,5)), but
- 1 times Hill recurrence [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]

(3, 6, 780)-Net in Base 5 — Upper bound on s

There is no (3, 6, 781)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁶, 781, S₅, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]