MinT - Best Known (28−8, 28, s)-Nets in Base 2

Best Known (28−8, 28, s)-Nets in Base 2

(28−8, 28, 53)-Net over F₂ — Constructive and digital

Digital (20, 28, 53)-net over F₂, using

net defined by OOA [i] based on linear OOA(2²⁸, 53, F₂, 8, 8) (dual of [(53, 8), 396, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2²⁸, 53, F₂, 7, 8) (dual of [(53, 7), 343, 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(2¹⁹, 30, F₂, 7, 8) (dual of [(30, 7), 191, 9]-NRT-code), using
      - extracting embedded OOA [i] based on digital (11, 19, 30)-net over F₂, using
        linear code embedding found in a computer search [i]

(28−8, 28, 64)-Net over F₂ — Digital

Digital (20, 28, 64)-net over F₂, using

net defined by OOA [i] based on linear OOA(2²⁸, 64, F₂, 8, 8) (dual of [(64, 8), 484, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2²⁸, 64, F₂, 7, 8) (dual of [(64, 7), 420, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁸, 64, F₂, 2, 8) (dual of [(64, 2), 100, 9]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2²⁸, 128, F₂, 8) (dual of [128, 100, 9]-code), using
      - 1 times truncation [i] based on linear OA(2²⁹, 129, F₂, 9) (dual of [129, 100, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 129Â | 2¹⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(28−8, 28, 277)-Net in Base 2 — Upper bound on s

There is no (20, 28, 278)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 268 986129 > 2²⁸ [i]