MinT - Best Known (107−8, 107, s)-Nets in Base 2

Best Known (107−8, 107, s)-Nets in Base 2

(107−8, 107, 2097299)-Net over F₂ — Constructive and digital

Digital (99, 107, 2097299)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰⁷, 2097299, F₂, 8, 8) (dual of [(2097299, 8), 16778285, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰⁷, 2097299, F₂, 7, 8) (dual of [(2097299, 7), 14680986, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2¹⁵, 149, F₂, 7, 4) (dual of [(149, 7), 1028, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2¹⁵, 149, F₂, 4, 4) (dual of [(149, 4), 581, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2¹⁵, 149, F₂, 3, 4) (dual of [(149, 3), 432, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (11, 15, 149)-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]

(107−8, 107, 4194452)-Net over F₂ — Digital

Digital (99, 107, 4194452)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁰⁷, 4194452, F₂, 8, 8) (dual of [(4194452, 8), 33555509, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁰⁷, 4194452, F₂, 7, 8) (dual of [(4194452, 7), 29361057, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁰⁷, 4194452, F₂, 2, 8) (dual of [(4194452, 2), 8388797, 9]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2¹⁵, 151, F₂, 2, 4) (dual of [(151, 2), 287, 5]-NRT-code), using
        embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(2¹⁵, 151, F₂, 4) (dual of [151, 136, 5]-code), using
        a â€œGBâ€ code from Brouwerâ€™s database [i]
      2. 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]

(107−8, 107, large)-Net in Base 2 — Upper bound on s

There is no (99, 107, large)-net in base 2, because

6 times m-reduction [i] would yield (99, 101, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]