MinT - Best Known (99−9, 99, s)-Nets in Base 2

Best Known (99−9, 99, s)-Nets in Base 2

(99−9, 99, 2097158)-Net over F₂ — Constructive and digital

Digital (90, 99, 2097158)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁹⁹, 2097158, F₂, 9, 9) (dual of [(2097158, 9), 18874323, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁹⁹, 2097158, F₂, 8, 9) (dual of [(2097158, 8), 16777165, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁶, 8, F₂, 8, 4) (dual of [(8, 8), 58, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2⁶, 8, F₂, 4, 4) (dual of [(8, 4), 26, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁶, 8, F₂, 3, 4) (dual of [(8, 3), 18, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (2, 6, 8)-net over F₂, using
        hand-crafted matrix by Schmid and Wolf [i]
    2. linear OOA(2⁹³, 2097150, F₂, 8, 9) (dual of [(2097150, 8), 16777107, 10]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OA(2⁹³, 8388601, F₂, 9) (dual of [8388601, 8388508, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(2⁹³, large, F₂, 9) (dual of [large, large−93, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(99−9, 99, large)-Net in Base 2 — Upper bound on s

There is no (90, 99, large)-net in base 2, because

7 times m-reduction [i] would yield (90, 92, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]