MinT - Best Known (24−5, 24, s)-Nets in Base 9

Best Known (24−5, 24, s)-Nets in Base 9

(24−5, 24, 58418)-Net over F₉ — Constructive and digital

Digital (19, 24, 58418)-net over F₉, using

net defined by OOA [i] based on linear OOA(9²⁴, 58418, F₉, 6, 5) (dual of [(58418, 6), 350484, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(9²⁴, 58419, F₉, 2, 5) (dual of [(58419, 2), 116814, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(9⁰, 6491, F₉, 2, 0) (dual of [(6491, 2), 12982, 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, 6491, F₉, 2, 0) (dual of [(6491, 2), 12982, 1]-NRT-code) (see above)
    3. linear OOA(9⁰, 6491, F₉, 2, 0) (dual of [(6491, 2), 12982, 1]-NRT-code) (see above)
    4. linear OOA(9⁰, 6491, F₉, 2, 0) (dual of [(6491, 2), 12982, 1]-NRT-code) (see above)
    5. linear OOA(9¹, 6491, F₉, 2, 1) (dual of [(6491, 2), 12981, 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, 6491, F₉, 2, 1) (dual of [(6491, 2), 12981, 2]-NRT-code) (see above)
    7. linear OOA(9¹, 6491, F₉, 2, 1) (dual of [(6491, 2), 12981, 2]-NRT-code) (see above)
    8. linear OOA(9⁵, 6491, F₉, 2, 2) (dual of [(6491, 2), 12977, 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(9¹⁶, 6491, F₉, 2, 5) (dual of [(6491, 2), 12966, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(9², 10, F₉, 2, 2) (dual of [(10, 2), 18, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(2;18,9) [i]
        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]

(24−5, 24, 147036)-Net over F₉ — Digital

Digital (19, 24, 147036)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(24−5, 24, large)-Net in Base 9 — Upper bound on s

There is no (19, 24, large)-net in base 9, because

3 times m-reduction [i] would yield (19, 21, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]

Best Known (24−5, 24, s)-Nets in Base 9

(24−5, 24, 58418)-Net over F9 — Constructive and digital

(24−5, 24, 147036)-Net over F9 — Digital

(24−5, 24, large)-Net in Base 9 — Upper bound on s

(24−5, 24, 58418)-Net over F₉ — Constructive and digital

(24−5, 24, 147036)-Net over F₉ — Digital