MinT - Best Known (5−3, 5, s)-Nets in Base 81

Best Known (5−3, 5, s)-Nets in Base 81

(5−3, 5, 16318)-Net over F₈₁ — Constructive and digital

Digital (2, 5, 16318)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81⁵, 16318, F₈₁, 3, 3) (dual of [(16318, 3), 48949, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(81⁵, 16318, F₈₁, 2, 3) (dual of [(16318, 2), 32631, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(81⁵, 16318, F₈₁, 3) (dual of [16318, 16313, 4]-code or 16318-cap in PG(4,81)), using
    - large caps in PG(4,Â b) [i]

(5−3, 5, 525514)-Net over F₈₁ — Upper bound on s (digital)

There is no digital (2, 5, 525515)-net over F₈₁, because

extracting embedded orthogonal array [i] would yield linear OA(81⁵, 525515, F₈₁, 3) (dual of [525515, 525510, 4]-code or 525515-cap in PG(4,81)), but
- bound for caps in PG(4,Â b) [i]

(5−3, 5, 538083)-Net in Base 81 — Upper bound on s

There is no (2, 5, 538084)-net in base 81, because

extracting embedded orthogonal array [i] would yield OA(81⁵, 538084, S₈₁, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]