MinT - Best Known (88−8, 88, s)-Nets in Base 2

Best Known (88−8, 88, s)-Nets in Base 2

(88−8, 88, 1048576)-Net over F₂ — Constructive and digital

Digital (80, 88, 1048576)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁸⁸, 1048576, F₂, 8, 8) (dual of [(1048576, 8), 8388520, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2⁸⁸, 4194304, F₂, 8) (dual of [4194304, 4194216, 9]-code), using
  - 1 times truncation [i] based on linear OA(2⁸⁹, 4194305, F₂, 9) (dual of [4194305, 4194216, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 2⁴⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(88−8, 88, 1398101)-Net over F₂ — Digital

Digital (80, 88, 1398101)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁸⁸, 1398101, F₂, 8, 8) (dual of [(1398101, 8), 11184720, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁸⁸, 1398101, F₂, 7, 8) (dual of [(1398101, 7), 9786619, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁸⁸, 1398101, F₂, 3, 8) (dual of [(1398101, 3), 4194215, 9]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2⁸⁸, 4194303, F₂, 8) (dual of [4194303, 4194215, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(2⁸⁸, 4194304, F₂, 8) (dual of [4194304, 4194216, 9]-code), using
        1 times truncation [i] based on linear OA(2⁸⁹, 4194305, F₂, 9) (dual of [4194305, 4194216, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 2⁴⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(88−8, 88, large)-Net in Base 2 — Upper bound on s

There is no (80, 88, large)-net in base 2, because

6 times m-reduction [i] would yield (80, 82, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]