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

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

(3, 3+3, 56)-Net over F₃ — Constructive and digital

Digital (3, 6, 56)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁶, 56, F₃, 3, 3) (dual of [(56, 3), 162, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁶, 56, F₃, 2, 3) (dual of [(56, 2), 106, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁶, 56, F₃, 3) (dual of [56, 50, 4]-code or 56-cap in PG(5,3)), using
    - the Hill cap [i]

(3, 3+3, 56)-Net over F₃ — Upper bound on s (digital)

There is no digital (3, 6, 57)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁶, 57, F₃, 3) (dual of [57, 51, 4]-code or 57-cap in PG(5,3)), but
- doubling the cap [i] would yield 114-cap in AG(6,3), but
  - discarding factors / shortening the dual code would yield 113-cap in AG(6,3), but
    - bound on the size of affine caps [i]

(3, 3+3, 120)-Net in Base 3 — Upper bound on s

There is no (3, 6, 121)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁶, 121, S₃, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]