MinT - Best Known (99−15, 99, s)-Nets in Base 2

Best Known (99−15, 99, s)-Nets in Base 2

Digital (84, 99, 2340)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁹⁹, 2340, F₂, 15, 15) (dual of [(2340, 15), 35001, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2⁹⁹, 16381, F₂, 15) (dual of [16381, 16282, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(2⁹⁹, 16384, F₂, 15) (dual of [16384, 16285, 16]-code), using
    - an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 2¹⁴âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

Digital (84, 99, 3276)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁹⁹, 3276, F₂, 5, 15) (dual of [(3276, 5), 16281, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2⁹⁹, 16380, F₂, 15) (dual of [16380, 16281, 16]-code), using
  - discarding factors / shortening the dual code based on linear OA(2⁹⁹, 16384, F₂, 15) (dual of [16384, 16285, 16]-code), using
    - an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 2¹⁴âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

There is no (84, 99, 55369)-net in base 2, because

1 times m-reduction [i] would yield (84, 98, 55369)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 316945 934018 994744 324171 457280 > 2⁹⁸ [i]