MinT - Best Known (115−13, 115, s)-Nets in Base 2

Best Known (115−13, 115, s)-Nets in Base 2

Digital (102, 115, 87381)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹¹⁵, 87381, F₂, 13, 13) (dual of [(87381, 13), 1135838, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2¹¹⁵, 524287, F₂, 13) (dual of [524287, 524172, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹¹⁵, 524288, F₂, 13) (dual of [524288, 524173, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 524287Â = 2¹⁹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

Digital (102, 115, 104857)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹⁵, 104857, F₂, 5, 13) (dual of [(104857, 5), 524170, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2¹¹⁵, 524285, F₂, 13) (dual of [524285, 524170, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹¹⁵, 524288, F₂, 13) (dual of [524288, 524173, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 524287Â = 2¹⁹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

There is no (102, 115, 1569603)-net in base 2, because

1 times m-reduction [i] would yield (102, 114, 1569603)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 20769 236676 332356 056484 013855 397568 > 2¹¹⁴ [i]