MinT - Best Known (2, 2+3, s)-Nets in Base 27

Best Known (2, 2+3, s)-Nets in Base 27

(2, 2+3, 1708)-Net over F₂₇ — Constructive and digital

Digital (2, 5, 1708)-net over F₂₇, using

net defined by OOA [i] based on linear OOA(27⁵, 1708, F₂₇, 3, 3) (dual of [(1708, 3), 5119, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(27⁵, 1708, F₂₇, 2, 3) (dual of [(1708, 2), 3411, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁵, 1708, F₂₇, 3) (dual of [1708, 1703, 4]-code or 1708-cap in PG(4,27)), using
    - large caps in PG(4,Â b) [i]

(2, 2+3, 19156)-Net over F₂₇ — Upper bound on s (digital)

There is no digital (2, 5, 19157)-net over F₂₇, because

extracting embedded orthogonal array [i] would yield linear OA(27⁵, 19157, F₂₇, 3) (dual of [19157, 19152, 4]-code or 19157-cap in PG(4,27)), but
- bound for caps in PG(4,Â b) [i]

(2, 2+3, 20439)-Net in Base 27 — Upper bound on s

There is no (2, 5, 20440)-net in base 27, because

extracting embedded orthogonal array [i] would yield OA(27⁵, 20440, S₂₇, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]