MinT - Best Known (5−3, 5, s)-Nets in Base 7

Best Known (5−3, 5, s)-Nets in Base 7

(5−3, 5, 132)-Net over F₇ — Constructive and digital

Digital (2, 5, 132)-net over F₇, using

net defined by OOA [i] based on linear OOA(7⁵, 132, F₇, 3, 3) (dual of [(132, 3), 391, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(7⁵, 132, F₇, 2, 3) (dual of [(132, 2), 259, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7⁵, 132, F₇, 3) (dual of [132, 127, 4]-code or 132-cap in PG(4,7)), using
    - caps completed using a computer search [i]

(5−3, 5, 238)-Net over F₇ — Upper bound on s (digital)

There is no digital (2, 5, 239)-net over F₇, because

extracting embedded orthogonal array [i] would yield linear OA(7⁵, 239, F₇, 3) (dual of [239, 234, 4]-code or 239-cap in PG(4,7)), but
- a result by Edel, Storme, and Sziklai [i]

(5−3, 5, 399)-Net in Base 7 — Upper bound on s

There is no (2, 5, 400)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7⁵, 400, S₇, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]