MinT - Best Known (104−8, 104, s)-Nets in Base 2

Best Known (104−8, 104, s)-Nets in Base 2

(104−8, 104, 2097215)-Net over F₂ — Constructive and digital

Digital (96, 104, 2097215)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰⁴, 2097215, F₂, 8, 8) (dual of [(2097215, 8), 16777616, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰⁴, 2097215, F₂, 7, 8) (dual of [(2097215, 7), 14680401, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2¹², 65, F₂, 7, 4) (dual of [(65, 7), 443, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2¹², 65, F₂, 4, 4) (dual of [(65, 4), 248, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2¹², 65, F₂, 3, 4) (dual of [(65, 3), 183, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (8, 12, 65)-net over F₂, using
        nets constructed from net-embeddable BCH codes [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]

(104−8, 104, 3604519)-Net over F₂ — Digital

Digital (96, 104, 3604519)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰⁴, 3604519, F₂, 8, 8) (dual of [(3604519, 8), 28836048, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰⁴, 3604519, F₂, 7, 8) (dual of [(3604519, 7), 25231529, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁰⁴, 3604519, F₂, 2, 8) (dual of [(3604519, 2), 7208934, 9]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(2¹⁰⁴, 4194366, F₂, 2, 8) (dual of [(4194366, 2), 8388628, 9]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2¹², 65, F₂, 2, 4) (dual of [(65, 2), 118, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (8, 12, 65)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
        linear OOA(2⁹², 4194301, F₂, 2, 8) (dual of [(4194301, 2), 8388510, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(2⁹², 8388602, F₂, 8) (dual of [8388602, 8388510, 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]

(104−8, 104, large)-Net in Base 2 — Upper bound on s

There is no (96, 104, large)-net in base 2, because

6 times m-reduction [i] would yield (96, 98, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]