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

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

(5, 5+3, 248)-Net over F₃ — Constructive and digital

Digital (5, 8, 248)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁸, 248, F₃, 3, 3) (dual of [(248, 3), 736, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁸, 248, F₃, 2, 3) (dual of [(248, 2), 488, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁸, 248, F₃, 3) (dual of [248, 240, 4]-code or 248-cap in PG(7,3)), using
    - large caps in PG(u,Â 3) [i]

(5, 5+3, 386)-Net over F₃ — Upper bound on s (digital)

There is no digital (5, 8, 387)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁸, 387, F₃, 3) (dual of [387, 379, 4]-code or 387-cap in PG(7,3)), but
- doubling the cap [i] would yield 774-cap in AG(8,3), but
  - 2 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]

(5, 5+3, 1092)-Net in Base 3 — Upper bound on s

There is no (5, 8, 1093)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸, 1093, S₃, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]