MinT - Best Known (22−5, 22, s)-Nets in Base 9

Best Known (22−5, 22, s)-Nets in Base 9

(22−5, 22, 58328)-Net over F₉ — Constructive and digital

Digital (17, 22, 58328)-net over F₉, using

net defined by OOA [i] based on linear OOA(9²², 58328, F₉, 6, 5) (dual of [(58328, 6), 349946, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(9²², 58329, F₉, 2, 5) (dual of [(58329, 2), 116636, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(9⁰, 6481, F₉, 2, 0) (dual of [(6481, 2), 12962, 1]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9⁰, s, F₉, 2, 0) with arbitrarily large s, using
        OOA with only one run / dual of NRT-code without redundancy [i]
    2. linear OOA(9⁰, 6481, F₉, 2, 0) (dual of [(6481, 2), 12962, 1]-NRT-code) (see above)
    3. linear OOA(9⁰, 6481, F₉, 2, 0) (dual of [(6481, 2), 12962, 1]-NRT-code) (see above)
    4. linear OOA(9⁰, 6481, F₉, 2, 0) (dual of [(6481, 2), 12962, 1]-NRT-code) (see above)
    5. linear OOA(9¹, 6481, F₉, 2, 1) (dual of [(6481, 2), 12961, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9¹, s, F₉, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    6. linear OOA(9¹, 6481, F₉, 2, 1) (dual of [(6481, 2), 12961, 2]-NRT-code) (see above)
    7. linear OOA(9¹, 6481, F₉, 2, 1) (dual of [(6481, 2), 12961, 2]-NRT-code) (see above)
    8. linear OOA(9⁵, 6481, F₉, 2, 2) (dual of [(6481, 2), 12957, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(9⁵, 7381, F₉, 2, 2) (dual of [(7381, 2), 14757, 3]-NRT-code), using
        appending kth column [i] based on linear OA(9⁵, 7381, F₉, 2) (dual of [7381, 7376, 3]-code), using
        Hamming code H(5,9) [i]
    9. linear OOA(9¹⁴, 6481, F₉, 2, 5) (dual of [(6481, 2), 12948, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(9¹⁴, 12962, F₉, 5) (dual of [12962, 12948, 6]-code), using
        trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(22−5, 22, 72010)-Net over F₉ — Digital

Digital (17, 22, 72010)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9²², 72010, F₉, 5) (dual of [72010, 71988, 6]-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OA(9⁰, 7381, F₉, 0) (dual of [7381, 7381, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]
  2. linear OA(9⁰, 7381, F₉, 0) (dual of [7381, 7381, 1]-code) (see above)
  3. linear OA(9⁰, 7381, F₉, 0) (dual of [7381, 7381, 1]-code) (see above)
  4. linear OA(9⁰, 7381, F₉, 0) (dual of [7381, 7381, 1]-code) (see above)
  5. linear OA(9¹, 7381, F₉, 1) (dual of [7381, 7380, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(9¹, s, F₉, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]
  6. linear OA(9¹, 7381, F₉, 1) (dual of [7381, 7380, 2]-code) (see above)
  7. linear OA(9¹, 7381, F₉, 1) (dual of [7381, 7380, 2]-code) (see above)
  8. linear OA(9⁵, 7381, F₉, 2) (dual of [7381, 7376, 3]-code), using
    - Hamming code H(5,9) [i]
  9. linear OA(9¹⁴, 12962, F₉, 5) (dual of [12962, 12948, 6]-code), using
    - trace code [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
      - a code by Danev and Olsson [i]

(22−5, 22, large)-Net in Base 9 — Upper bound on s

There is no (17, 22, large)-net in base 9, because

3 times m-reduction [i] would yield (17, 19, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (22−5, 22, s)-Nets in Base 9

(22−5, 22, 58328)-Net over F9 — Constructive and digital

(22−5, 22, 72010)-Net over F9 — Digital

(22−5, 22, large)-Net in Base 9 — Upper bound on s

(22−5, 22, 58328)-Net over F₉ — Constructive and digital

(22−5, 22, 72010)-Net over F₉ — Digital