MinT - Best Known (30−3, 30, s)-Nets in Base 7

Best Known (30−3, 30, s)-Nets in Base 7

(30−3, 30, large)-Net over F₇ — Constructive and digital

Digital (27, 30, large)-net over F₇, using

7¹⁶ times duplication [i] based on digital (11, 14, large)-net over F₇, using
- net defined by OOA [i] based on linear OOA(7¹⁴, large, F₇, 3, 3), using
  - appending kth column [i] based on linear OOA(7¹⁴, large, F₇, 2, 3), using
    - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7¹⁴, large, F₇, 3) (dual of [large, large−14, 4]-code), using
      - doubling the cap [i] based on linear OA(7¹³, 6120101, F₇, 3) (dual of [6120101, 6120088, 4]-code or 6120101-cap in PG(12,7)), using
        product of projective cap with cap in AG(6,Â 7) [i] based on linear OA(7⁷, 2499, F₇, 3) (dual of [2499, 2492, 4]-code or 2499-cap in PG(6,7)), using
        product of two caps with tangent hyperplane [i]

(30−3, 30, large)-Net in Base 7 — Upper bound on s

There is no (27, 30, large)-net in base 7, because

1 times m-reduction [i] would yield (27, 29, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]