MinT - Best Known (4, 4+3, s)-Nets in Base 5

Best Known (4, 4+3, s)-Nets in Base 5

(4, 4+3, 675)-Net over F₅ — Constructive and digital

Digital (4, 7, 675)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁷, 675, F₅, 3, 3) (dual of [(675, 3), 2018, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(5⁷, 675, F₅, 2, 3) (dual of [(675, 2), 1343, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5⁷, 675, F₅, 3) (dual of [675, 668, 4]-code or 675-cap in PG(6,5)), using
    - product of two caps with tangent hyperplane [i]

(4, 4+3, 2152)-Net over F₅ — Upper bound on s (digital)

There is no digital (4, 7, 2153)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5⁷, 2153, F₅, 3) (dual of [2153, 2146, 4]-code or 2153-cap in PG(6,5)), 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)

(4, 4+3, 3905)-Net in Base 5 — Upper bound on s

There is no (4, 7, 3906)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁷, 3906, S₅, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]