MinT - Best Known (4, 4+3, s)-Nets in Base 8

Best Known (4, 4+3, s)-Nets in Base 8

(4, 4+3, 4224)-Net over F₈ — Constructive and digital

Digital (4, 7, 4224)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁷, 4224, F₈, 3, 3) (dual of [(4224, 3), 12665, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(8⁷, 4224, F₈, 2, 3) (dual of [(4224, 2), 8441, 4]-NRT-code), using
  - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8⁷, 4224, F₈, 3) (dual of [4224, 4217, 4]-code or 4224-cap in PG(6,8)), using
    - product of two caps with tangent hyperplane [i]

(4, 4+3, 27649)-Net over F₈ — Upper bound on s (digital)

There is no digital (4, 7, 27650)-net over F₈, because

extracting embedded orthogonal array [i] would yield linear OA(8⁷, 27650, F₈, 3) (dual of [27650, 27643, 4]-code or 27650-cap in PG(6,8)), but
- removing affine subspaces [i] would yield
  1. linear OA(8⁵, 494, F₈, 3) (dual of [494, 489, 4]-code or 494-cap in PG(4,8)), but
    - bound for caps in PG(4,Â b) [i]
  2. 3284-cap in AG(5,8), but
    - 2 times the recursive bound from Bierbrauer and Edel [i] would yield 65-cap in AG(3,8), but
      - affine subcap of ovoid is optimal [i]
  3. 23874-cap in AG(6,8), but
    - 3 times the recursive bound from Bierbrauer and Edel [i] would yield 65-cap in AG(3,8) (see above)

(4, 4+3, 37448)-Net in Base 8 — Upper bound on s

There is no (4, 7, 37449)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8⁷, 37449, S₈, 3), but
- Boseâ€“Bush bound for OAs with strength 3 [i]