MinT - Best Known (5, 8, s)-Nets in Base 5

Best Known (5, 8, s)-Nets in Base 5

(5, 8, 1715)-Net over F₅ — Constructive and digital

Digital (5, 8, 1715)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁸, 1715, F₅, 3, 3) (dual of [(1715, 3), 5137, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(5⁸, 1715, F₅, 2, 3) (dual of [(1715, 2), 3422, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁸, 1715, F₅, 3) (dual of [1715, 1707, 4]-code or 1715-cap in PG(7,5)), using
    - product of two caps with tangent hyperplane [i]

(5, 8, 9891)-Net over F₅ — Upper bound on s (digital)

There is no digital (5, 8, 9892)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5⁸, 9892, F₅, 3) (dual of [9892, 9884, 4]-code or 9892-cap in PG(7,5)), but
- removing affine subspaces [i] would yield
  1. linear OA(5⁶, 435, F₅, 3) (dual of [435, 429, 4]-code or 435-cap in PG(5,5)), but
    - 1 times Hill recurrence [i] would yield linear OA(5⁵, 89, F₅, 3) (dual of [89, 84, 4]-code or 89-cap in PG(4,5)), but
      - a result by Edel, Storme, and Sziklai [i]
  2. 1719-cap in AG(6,5), but
    - 2 times the recursive bound from Bierbrauer and Edel [i] would yield 89-cap in AG(4,5), but
      - embedding in projective space [i] would yield linear OA(5⁵, 89, F₅, 3) (dual of [89, 84, 4]-code or 89-cap in PG(4,5)) (see above)
  3. 7740-cap in AG(7,5), but
    - 3 times the recursive bound from Bierbrauer and Edel [i] would yield 89-cap in AG(4,5) (see above)

(5, 8, 19530)-Net in Base 5 — Upper bound on s

There is no (5, 8, 19531)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁸, 19531, S₅, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]