MinT - Best Known (135−23, 135, s)-Nets in Base 3

Best Known (135−23, 135, s)-Nets in Base 3

Digital (112, 135, 894)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹³⁵, 894, F₃, 23, 23) (dual of [(894, 23), 20427, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3¹³⁵, 9835, F₃, 23) (dual of [9835, 9700, 24]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹³⁵, 9841, F₃, 23) (dual of [9841, 9706, 24]-code), using
    - the BCH-code C(I) with length 9841Â | 3⁹âˆ’1, defining interval IÂ =Â {1,3,5,…,45}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

Digital (112, 135, 4920)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³⁵, 4920, F₃, 2, 23) (dual of [(4920, 2), 9705, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹³⁵, 9840, F₃, 23) (dual of [9840, 9705, 24]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹³⁵, 9841, F₃, 23) (dual of [9841, 9706, 24]-code), using
    - the BCH-code C(I) with length 9841Â | 3⁹âˆ’1, defining interval IÂ =Â {1,3,5,…,45}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

There is no (112, 135, 1592890)-net in base 3, because

1 times m-reduction [i] would yield (112, 134, 1592890)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8595 069022 911964 615769 735662 555380 922978 732900 469882 050604 033697 > 3¹³⁴ [i]