MinT - Best Known (92−22, 92, s)-Nets in Base 9

Best Known (92−22, 92, s)-Nets in Base 9

(92−22, 92, 1194)-Net over F₉ — Constructive and digital

Digital (70, 92, 1194)-net over F₉, using

1 times m-reduction [i] based on digital (70, 93, 1194)-net over F₉, using
- net defined by OOA [i] based on linear OOA(9⁹³, 1194, F₉, 23, 23) (dual of [(1194, 23), 27369, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(9⁹³, 13135, F₉, 23) (dual of [13135, 13042, 24]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(9⁹², 13134, F₉, 23) (dual of [13134, 13042, 24]-code), using
      - trace code [i] based on linear OA(81⁴⁶, 6567, F₈₁, 23) (dual of [6567, 6521, 24]-code), using
        constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
        linear OA(81⁴⁵, 6562, F₈₁, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(81⁴¹, 6562, F₈₁, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(81¹, 5, F₈₁, 1) (dual of [5, 4, 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]

(92−22, 92, 16450)-Net over F₉ — Digital

Digital (70, 92, 16450)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(92−22, 92, large)-Net in Base 9 — Upper bound on s

There is no (70, 92, large)-net in base 9, because

20 times m-reduction [i] would yield (70, 72, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]