MinT - Best Known (6, 11, s)-Nets in Base 81

Best Known (6, 11, s)-Nets in Base 81

(6, 11, 9721)-Net over F₈₁ — Constructive and digital

Digital (6, 11, 9721)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81¹¹, 9721, F₈₁, 5, 5) (dual of [(9721, 5), 48594, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(81¹¹, 9721, F₈₁, 4, 5) (dual of [(9721, 4), 38873, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(81¹¹, 19443, F₈₁, 5) (dual of [19443, 19432, 6]-code), using
    - (u,Â uâˆ’v,Â u+v+w)-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(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]
      3. linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(6, 11, 19443)-Net over F₈₁ — Digital

Digital (6, 11, 19443)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81¹¹, 19443, F₈₁, 5, 5) (dual of [(19443, 5), 97204, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(81¹¹, 19443, F₈₁, 4, 5) (dual of [(19443, 4), 77761, 6]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(81¹¹, 19443, F₈₁, 5) (dual of [19443, 19432, 6]-code), using
    - (u,Â uâˆ’v,Â u+v+w)-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(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]
      3. linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(6, 11, large)-Net in Base 81 — Upper bound on s

There is no (6, 11, large)-net in base 81, because

3 times m-reduction [i] would yield (6, 8, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]