MinT - Best Known (7, 12, s)-Nets in Base 81

Best Known (7, 12, s)-Nets in Base 81

(7, 12, 12961)-Net over F₈₁ — Constructive and digital

Digital (7, 12, 12961)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81¹², 12961, F₈₁, 5, 5) (dual of [(12961, 5), 64793, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(81¹², 12961, F₈₁, 4, 5) (dual of [(12961, 4), 51832, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(81¹², 25923, F₈₁, 5) (dual of [25923, 25911, 6]-code), using
    - discarding factors / shortening the dual code based on linear OA(81¹², 25924, F₈₁, 5) (dual of [25924, 25912, 6]-code), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OA(81¹, 6481, F₈₁, 1) (dual of [6481, 6480, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OA(81¹, 6481, F₈₁, 1) (dual of [6481, 6480, 2]-code) (see above)
        linear OA(81³, 6481, F₈₁, 2) (dual of [6481, 6478, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(81³, 6643, F₈₁, 2) (dual of [6643, 6640, 3]-code), using
        Hamming code H(3,81) [i]
        linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(7, 12, 25924)-Net over F₈₁ — Digital

Digital (7, 12, 25924)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81¹², 25924, F₈₁, 5, 5) (dual of [(25924, 5), 129608, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(81¹², 25924, F₈₁, 4, 5) (dual of [(25924, 4), 103684, 6]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81¹², 25924, F₈₁, 5) (dual of [25924, 25912, 6]-code), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OA(81¹, 6481, F₈₁, 1) (dual of [6481, 6480, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(81¹, s, F₈₁, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
      2. linear OA(81¹, 6481, F₈₁, 1) (dual of [6481, 6480, 2]-code) (see above)
      3. linear OA(81³, 6481, F₈₁, 2) (dual of [6481, 6478, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(81³, 6643, F₈₁, 2) (dual of [6643, 6640, 3]-code), using
        Hamming code H(3,81) [i]
      4. linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(7, 12, large)-Net in Base 81 — Upper bound on s

There is no (7, 12, large)-net in base 81, because

3 times m-reduction [i] would yield (7, 9, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]