MinT - Best Known (9, 12, s)-Nets in Base 49

Best Known (9, 12, s)-Nets in Base 49

(9, 12, large)-Net over F₄₉ — Constructive and digital

Digital (9, 12, large)-net over F₄₉, using

49⁴ times duplication [i] based on digital (5, 8, large)-net over F₄₉, using
- net defined by OOA [i] based on linear OOA(49⁸, large, F₄₉, 3, 3), using
  - appending kth column [i] based on linear OOA(49⁸, large, F₄₉, 2, 3), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(49⁸, large, F₄₉, 3) (dual of [large, large−8, 4]-code), using
      - doubling the cap [i] based on linear OA(49⁷, 5769603, F₄₉, 3) (dual of [5769603, 5769596, 4]-code or 5769603-cap in PG(6,49)), using
        product of two caps with tangent hyperplane [i]

(9, 12, large)-Net in Base 49 — Upper bound on s

There is no (9, 12, large)-net in base 49, because

1 times m-reduction [i] would yield (9, 11, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]