MinT - Best Known (106−27, 106, s)-Nets in Base 9

Best Known (106−27, 106, s)-Nets in Base 9

Digital (79, 106, 1009)-net over F₉, using

net defined by OOA [i] based on linear OOA(9¹⁰⁶, 1009, F₉, 27, 27) (dual of [(1009, 27), 27137, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9¹⁰⁶, 13118, F₉, 27) (dual of [13118, 13012, 28]-code), using
  - discarding factors / shortening the dual code based on linear OA(9¹⁰⁶, 13124, F₉, 27) (dual of [13124, 13018, 28]-code), using
    - trace code [i] based on linear OA(81⁵³, 6562, F₈₁, 27) (dual of [6562, 6509, 28]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]

Digital (79, 106, 12938)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9¹⁰⁶, 12938, F₉, 27) (dual of [12938, 12832, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9¹⁰⁶, 13124, F₉, 27) (dual of [13124, 13018, 28]-code), using
  - trace code [i] based on linear OA(81⁵³, 6562, F₈₁, 27) (dual of [6562, 6509, 28]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]

There is no (79, 106, large)-net in base 9, because

25 times m-reduction [i] would yield (79, 81, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]