MinT - Best Known (5−3, 5, s)-Nets in Base 49

Best Known (5−3, 5, s)-Nets in Base 49

(5−3, 5, 5950)-Net over F₄₉ — Constructive and digital

Digital (2, 5, 5950)-net over F₄₉, using

net defined by OOA [i] based on linear OOA(49⁵, 5950, F₄₉, 3, 3) (dual of [(5950, 3), 17845, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(49⁵, 5950, F₄₉, 2, 3) (dual of [(5950, 2), 11895, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(49⁵, 5950, F₄₉, 3) (dual of [5950, 5945, 4]-code or 5950-cap in PG(4,49)), using
    - large caps in PG(4,Â b) [i]

(5−3, 5, 115626)-Net over F₄₉ — Upper bound on s (digital)

There is no digital (2, 5, 115627)-net over F₄₉, because

extracting embedded orthogonal array [i] would yield linear OA(49⁵, 115627, F₄₉, 3) (dual of [115627, 115622, 4]-code or 115627-cap in PG(4,49)), but
- bound for caps in PG(4,Â b) [i]

(5−3, 5, 120099)-Net in Base 49 — Upper bound on s

There is no (2, 5, 120100)-net in base 49, because

extracting embedded orthogonal array [i] would yield OA(49⁵, 120100, S₄₉, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]