MinT - Best Known (21−3, 21, s)-Nets in Base 8

Best Known (21−3, 21, s)-Nets in Base 8

(21−3, 21, large)-Net over F₈ — Constructive and digital

Digital (18, 21, large)-net over F₈, using

8⁸ times duplication [i] based on digital (10, 13, large)-net over F₈, using
- net defined by OOA [i] based on linear OOA(8¹³, large, F₈, 3, 3), using
  - appending kth column [i] based on linear OOA(8¹³, large, F₈, 2, 3), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(8¹³, large, F₈, 3) (dual of [large, large−13, 4]-code), using
      - product of projective cap with (hyper)oval [i] based on linear OA(8¹¹, 878592, F₈, 3) (dual of [878592, 878581, 4]-code or 878592-cap in PG(10,8)), using
        product of projective cap with cap in AG(4,Â 8) [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]

(21−3, 21, large)-Net in Base 8 — Upper bound on s

There is no (18, 21, large)-net in base 8, because

1 times m-reduction [i] would yield (18, 20, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]