MinT - Best Known (99−8, 99, s)-Nets in Base 2

Best Known (99−8, 99, s)-Nets in Base 2

(99−8, 99, 2097161)-Net over F₂ — Constructive and digital

Digital (91, 99, 2097161)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁹⁹, 2097161, F₂, 8, 8) (dual of [(2097161, 8), 16777189, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁹⁹, 2097161, F₂, 7, 8) (dual of [(2097161, 7), 14680028, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁷, 11, F₂, 7, 4) (dual of [(11, 7), 70, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2⁷, 11, F₂, 4, 4) (dual of [(11, 4), 37, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁷, 11, F₂, 3, 4) (dual of [(11, 3), 26, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (3, 7, 11)-net over F₂, using
        hand-crafted matrix by Schmid and Wolf [i]
    2. linear OOA(2⁹², 2097150, F₂, 7, 8) (dual of [(2097150, 7), 14679958, 9]-NRT-code), using
      - OA 4-folding and stacking [i] based on linear OA(2⁹², 8388600, F₂, 8) (dual of [8388600, 8388508, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(2⁹², large, F₂, 8) (dual of [large, large−92, 9]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(99−8, 99, 2796212)-Net over F₂ — Digital

Digital (91, 99, 2796212)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁹⁹, 2796212, F₂, 8, 8) (dual of [(2796212, 8), 22369597, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁹⁹, 2796212, F₂, 7, 8) (dual of [(2796212, 7), 19573385, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁹⁹, 2796212, F₂, 3, 8) (dual of [(2796212, 3), 8388537, 9]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁷, 11, F₂, 3, 4) (dual of [(11, 3), 26, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (3, 7, 11)-net over F₂, using
        hand-crafted matrix by Schmid and Wolf [i]
      2. linear OOA(2⁹², 2796201, F₂, 3, 8) (dual of [(2796201, 3), 8388511, 9]-NRT-code), using
        OOA 3-folding [i] based on linear OA(2⁹², large, F₂, 8) (dual of [large, large−92, 9]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(99−8, 99, large)-Net in Base 2 — Upper bound on s

There is no (91, 99, large)-net in base 2, because

6 times m-reduction [i] would yield (91, 93, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]