MinT - Best Known (18, 18+3, s)-Nets in Base 27

Best Known (18, 18+3, s)-Nets in Base 27

(18, 18+3, large)-Net over F₂₇ — Constructive and digital

Digital (18, 21, large)-net over F₂₇, using

27¹² times duplication [i] based on digital (6, 9, large)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27⁹, large, F₂₇, 3, 3), using
  - appending kth column [i] based on linear OOA(27⁹, large, F₂₇, 2, 3), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁹, large, F₂₇, 3) (dual of [large, large−9, 4]-code), using
      - product of projective cap with (hyper)oval [i] based on linear OA(27⁷, 532899, F₂₇, 3) (dual of [532899, 532892, 4]-code or 532899-cap in PG(6,27)), using
        product of two caps with tangent hyperplane [i]

(18, 18+3, large)-Net in Base 27 — Upper bound on s

There is no (18, 21, large)-net in base 27, because

1 times m-reduction [i] would yield (18, 20, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]