MinT - Best Known (18−3, 18, s)-Nets in Base 5

Best Known (18−3, 18, s)-Nets in Base 5

(18−3, 18, large)-Net over F₅ — Constructive and digital

Digital (15, 18, large)-net over F₅, using

5² times duplication [i] based on digital (13, 16, large)-net over F₅, using
- net defined by OOA [i] based on linear OOA(5¹⁶, large, F₅, 3, 3), using
  - appending kth column [i] based on linear OOA(5¹⁶, large, F₅, 2, 3), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5¹⁶, large, F₅, 3) (dual of [large, large−16, 4]-code), using
      - product of projective cap with 5²-ovoid cap in AG(3,Â 5) [i] based on linear OA(5¹³, 438101, F₅, 3) (dual of [438101, 438088, 4]-code or 438101-cap in PG(12,5)), using
        product of projective cap with cap in AG(6,Â 5) [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]

(18−3, 18, large)-Net in Base 5 — Upper bound on s

There is no (15, 18, large)-net in base 5, because

1 times m-reduction [i] would yield (15, 17, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]