MinT - Best Known (1, 1+3, s)-Nets in Base 128

Best Known (1, 1+3, s)-Nets in Base 128

(1, 1+3, 16385)-Net over F₁₂₈ — Constructive and digital

Digital (1, 4, 16385)-net over F₁₂₈, using

net defined by OOA [i] based on linear OOA(128⁴, 16385, F₁₂₈, 3, 3) (dual of [(16385, 3), 49151, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(128⁴, 16385, F₁₂₈, 2, 3) (dual of [(16385, 2), 32766, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(128⁴, 16385, F₁₂₈, 3) (dual of [16385, 16381, 4]-code or 16385-cap in PG(3,128)), using
    - ovoid in PG(3,Â 128) [i]

(1, 1+3, 16385)-Net over F₁₂₈ — Upper bound on s (digital)

There is no digital (1, 4, 16386)-net over F₁₂₈, because

extracting embedded orthogonal array [i] would yield linear OA(128⁴, 16386, F₁₂₈, 3) (dual of [16386, 16382, 4]-code or 16386-cap in PG(3,128)), but
- ovoid cap is optimal [i]

(1, 1+3, 16512)-Net in Base 128 — Upper bound on s

There is no (1, 4, 16513)-net in base 128, because

extracting embedded orthogonal array [i] would yield OA(128⁴, 16513, S₁₂₈, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]