MinT - Best Known (6, 9, s)-Nets in Base 3

Best Known (6, 9, s)-Nets in Base 3

(6, 9, 532)-Net over F₃ — Constructive and digital

Digital (6, 9, 532)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁹, 532, F₃, 3, 3) (dual of [(532, 3), 1587, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁹, 532, F₃, 2, 3) (dual of [(532, 2), 1055, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁹, 532, F₃, 3) (dual of [532, 523, 4]-code or 532-cap in PG(8,3)), using
    - caps completed using a computer search [i]

(6, 9, 1037)-Net over F₃ — Upper bound on s (digital)

There is no digital (6, 9, 1038)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁹, 1038, F₃, 3) (dual of [1038, 1029, 4]-code or 1038-cap in PG(8,3)), but
- doubling the cap [i] would yield 2076-cap in AG(9,3), but
  - 3 times the recursive bound from Bierbrauer and Edel [i] would yield 113-cap in AG(6,3), but
    - bound on the size of affine caps [i]

(6, 9, 3279)-Net in Base 3 — Upper bound on s

There is no (6, 9, 3280)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁹, 3280, S₃, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]