MinT - Best Known (93, 93+8, s)-Nets in Base 2

Best Known (93, 93+8, s)-Nets in Base 2

(93, 93+8, 2097173)-Net over F₂ — Constructive and digital

Digital (93, 101, 2097173)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰¹, 2097173, F₂, 8, 8) (dual of [(2097173, 8), 16777283, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰¹, 2097173, F₂, 7, 8) (dual of [(2097173, 7), 14680110, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁹, 23, F₂, 7, 4) (dual of [(23, 7), 152, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2⁹, 23, F₂, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁹, 23, F₂, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 9, 23)-net over F₂, using
        linear code embedding found in a computer search [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]

(93, 93+8, 2796224)-Net over F₂ — Digital

Digital (93, 101, 2796224)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰¹, 2796224, F₂, 8, 8) (dual of [(2796224, 8), 22369691, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰¹, 2796224, F₂, 7, 8) (dual of [(2796224, 7), 19573467, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁰¹, 2796224, F₂, 3, 8) (dual of [(2796224, 3), 8388571, 9]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁹, 23, F₂, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 9, 23)-net over F₂, using
        linear code embedding found in a computer search [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]

(93, 93+8, large)-Net in Base 2 — Upper bound on s

There is no (93, 101, large)-net in base 2, because

6 times m-reduction [i] would yield (93, 95, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]