MinT - Best Known (66−17, 66, s)-Nets in Base 9

Best Known (66−17, 66, s)-Nets in Base 9

Digital (49, 66, 1640)-net over F₉, using

net defined by OOA [i] based on linear OOA(9⁶⁶, 1640, F₉, 17, 17) (dual of [(1640, 17), 27814, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(9⁶⁶, 13121, F₉, 17) (dual of [13121, 13055, 18]-code), using
  - discarding factors / shortening the dual code based on linear OA(9⁶⁶, 13124, F₉, 17) (dual of [13124, 13058, 18]-code), using
    - trace code [i] based on linear OA(81³³, 6562, F₈₁, 17) (dual of [6562, 6529, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

Digital (49, 66, 10950)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁶⁶, 10950, F₉, 17) (dual of [10950, 10884, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(9⁶⁶, 13124, F₉, 17) (dual of [13124, 13058, 18]-code), using
  - trace code [i] based on linear OA(81³³, 6562, F₈₁, 17) (dual of [6562, 6529, 18]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 6562Â | 81⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

There is no (49, 66, large)-net in base 9, because

15 times m-reduction [i] would yield (49, 51, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]