MinT - Best Known (1, 4, s)-Nets in Base 8

Best Known (1, 4, s)-Nets in Base 8

(1, 4, 65)-Net over F₈ — Constructive and digital

Digital (1, 4, 65)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁴, 65, F₈, 3, 3) (dual of [(65, 3), 191, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(8⁴, 65, F₈, 2, 3) (dual of [(65, 2), 126, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁴, 65, F₈, 3) (dual of [65, 61, 4]-code or 65-cap in PG(3,8)), using
    - ovoid in PG(3,Â 8) [i]

(1, 4, 65)-Net over F₈ — Upper bound on s (digital)

There is no digital (1, 4, 66)-net over F₈, because

extracting embedded orthogonal array [i] would yield linear OA(8⁴, 66, F₈, 3) (dual of [66, 62, 4]-code or 66-cap in PG(3,8)), but
- ovoid cap is optimal [i]

(1, 4, 72)-Net in Base 8 — Upper bound on s

There is no (1, 4, 73)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8⁴, 73, S₈, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]