MinT - Best Known (22, 31, s)-Nets in Base 9

Best Known (22, 31, s)-Nets in Base 9

Digital (22, 31, 819)-net over F₉, using

net defined by OOA [i] based on linear OOA(9³¹, 819, F₉, 9, 9) (dual of [(819, 9), 7340, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(9³¹, 3277, F₉, 9) (dual of [3277, 3246, 10]-code), using
  - discarding factors / shortening the dual code based on linear OA(9³¹, 3280, F₉, 9) (dual of [3280, 3249, 10]-code), using
    - 1 times truncation [i] based on linear OA(9³², 3281, F₉, 10) (dual of [3281, 3249, 11]-code), using
      - the BCH-code C(I) with length 3281Â | 9⁸âˆ’1, defining interval IÂ =Â {−9,−7,−5,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

Digital (22, 31, 3280)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9³¹, 3280, F₉, 9) (dual of [3280, 3249, 10]-code), using
- 1 times truncation [i] based on linear OA(9³², 3281, F₉, 10) (dual of [3281, 3249, 11]-code), using
  - the BCH-code C(I) with length 3281Â | 9⁸âˆ’1, defining interval IÂ =Â {−9,−7,−5,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

There is no (22, 31, 3969917)-net in base 9, because

1 times m-reduction [i] would yield (22, 30, 3969917)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 42391 158869 314655 344988 343329 > 9³⁰ [i]