MinT - Best Known (90−15, 90, s)-Nets in Base 3

Best Known (90−15, 90, s)-Nets in Base 3

Digital (75, 90, 2811)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁹⁰, 2811, F₃, 15, 15) (dual of [(2811, 15), 42075, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3⁹⁰, 19678, F₃, 15) (dual of [19678, 19588, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(3⁹⁰, 19682, F₃, 15) (dual of [19682, 19592, 16]-code), using
    - 1 times truncation [i] based on linear OA(3⁹¹, 19683, F₃, 16) (dual of [19683, 19592, 17]-code), using
      - an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

Digital (75, 90, 8330)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁹⁰, 8330, F₃, 2, 15) (dual of [(8330, 2), 16570, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3⁹⁰, 9841, F₃, 2, 15) (dual of [(9841, 2), 19592, 16]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3⁹⁰, 19682, F₃, 15) (dual of [19682, 19592, 16]-code), using
    - 1 times truncation [i] based on linear OA(3⁹¹, 19683, F₃, 16) (dual of [19683, 19592, 17]-code), using
      - an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

There is no (75, 90, 1968536)-net in base 3, because

1 times m-reduction [i] would yield (75, 89, 1968536)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 909330 108170 074241 791859 153607 446148 835873 > 3⁸⁹ [i]