MinT - Best Known (2, 2+3, s)-Nets in Base 4

Best Known (2, 2+3, s)-Nets in Base 4

(2, 2+3, 41)-Net over F₄ — Constructive and digital

Digital (2, 5, 41)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁵, 41, F₄, 3, 3) (dual of [(41, 3), 118, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(4⁵, 41, F₄, 2, 3) (dual of [(41, 2), 77, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
    - a cap found by Tallinii [i]

(2, 2+3, 41)-Net over F₄ — Upper bound on s (digital)

There is no digital (2, 5, 42)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4⁵, 42, F₄, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,4)), but
- 41 is the largest size of a cap in PG(4,Â 4) [i]

(2, 2+3, 84)-Net in Base 4 — Upper bound on s

There is no (2, 5, 85)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵, 85, S₄, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]