MinT - Best Known (89, 97, s)-Nets in Base 2

Best Known (89, 97, s)-Nets in Base 2

(89, 97, 2097155)-Net over F₂ — Constructive and digital

Digital (89, 97, 2097155)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁹⁷, 2097155, F₂, 8, 8) (dual of [(2097155, 8), 16777143, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁹⁷, 2097155, F₂, 7, 8) (dual of [(2097155, 7), 14679988, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁵, 5, F₂, 7, 4) (dual of [(5, 7), 30, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2⁵, 5, F₂, 4, 4) (dual of [(5, 4), 15, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁵, 5, F₂, 3, 4) (dual of [(5, 3), 10, 5]-NRT-code), using
        extended algebraic-geometric NRT-code AG_e(3;F,10P) [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
        an explicitly constructive algebraic function field [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]

(89, 97, 2796206)-Net over F₂ — Digital

Digital (89, 97, 2796206)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁹⁷, 2796206, F₂, 8, 8) (dual of [(2796206, 8), 22369551, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁹⁷, 2796206, F₂, 7, 8) (dual of [(2796206, 7), 19573345, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁹⁷, 2796206, F₂, 3, 8) (dual of [(2796206, 3), 8388521, 9]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁵, 5, F₂, 3, 4) (dual of [(5, 3), 10, 5]-NRT-code), using
        extended algebraic-geometric NRT-code AG_e(3;F,10P) [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
        an explicitly constructive algebraic function field [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]

(89, 97, large)-Net in Base 2 — Upper bound on s

There is no (89, 97, large)-net in base 2, because

6 times m-reduction [i] would yield (89, 91, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]