MinT - Best Known (1, 4, s)-Nets in Base 32

Best Known (1, 4, s)-Nets in Base 32

(1, 4, 1025)-Net over F₃₂ — Constructive and digital

Digital (1, 4, 1025)-net over F₃₂, using

net defined by OOA [i] based on linear OOA(32⁴, 1025, F₃₂, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(32⁴, 1025, F₃₂, 2, 3) (dual of [(1025, 2), 2046, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁴, 1025, F₃₂, 3) (dual of [1025, 1021, 4]-code or 1025-cap in PG(3,32)), using
    - ovoid in PG(3,Â 32) [i]

(1, 4, 1025)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (1, 4, 1026)-net over F₃₂, because

extracting embedded orthogonal array [i] would yield linear OA(32⁴, 1026, F₃₂, 3) (dual of [1026, 1022, 4]-code or 1026-cap in PG(3,32)), but
- ovoid cap is optimal [i]

(1, 4, 1056)-Net in Base 32 — Upper bound on s

There is no (1, 4, 1057)-net in base 32, because

extracting embedded orthogonal array [i] would yield OA(32⁴, 1057, S₃₂, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]