MinT - Best Known (1, 1+3, s)-Nets in Base 9

Best Known (1, 1+3, s)-Nets in Base 9

(1, 1+3, 82)-Net over F₉ — Constructive and digital

Digital (1, 4, 82)-net over F₉, using

net defined by OOA [i] based on linear OOA(9⁴, 82, F₉, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(9⁴, 82, F₉, 2, 3) (dual of [(82, 2), 160, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(9⁴, 82, F₉, 3) (dual of [82, 78, 4]-code or 82-cap in PG(3,9)), using
    - ovoid in PG(3,Â 9) [i]

(1, 1+3, 82)-Net over F₉ — Upper bound on s (digital)

There is no digital (1, 4, 83)-net over F₉, because

extracting embedded orthogonal array [i] would yield linear OA(9⁴, 83, F₉, 3) (dual of [83, 79, 4]-code or 83-cap in PG(3,9)), but
- ovoid cap is optimal [i]

(1, 1+3, 90)-Net in Base 9 — Upper bound on s

There is no (1, 4, 91)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9⁴, 91, S₉, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]