MinT - Best Known (32−8, 32, s)-Nets in Base 2

Best Known (32−8, 32, s)-Nets in Base 2

(32−8, 32, 78)-Net over F₂ — Constructive and digital

Digital (24, 32, 78)-net over F₂, using

net defined by OOA [i] based on linear OOA(2³², 78, F₂, 8, 8) (dual of [(78, 8), 592, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2³², 78, F₂, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2¹¹, 47, F₂, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2¹¹, 47, F₂, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2¹¹, 47, F₂, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (7, 11, 47)-net over F₂, using
        linear code embedding found in a computer search [i]
    2. linear OOA(2²¹, 39, F₂, 7, 8) (dual of [(39, 7), 252, 9]-NRT-code), using
      - extracting embedded OOA [i] based on digital (13, 21, 39)-net over F₂, using
        linear code embedding found in a computer search [i]

(32−8, 32, 128)-Net over F₂ — Digital

Digital (24, 32, 128)-net over F₂, using

net defined by OOA [i] based on linear OOA(2³², 128, F₂, 8, 8) (dual of [(128, 8), 992, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2³², 128, F₂, 7, 8) (dual of [(128, 7), 864, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³², 128, F₂, 2, 8) (dual of [(128, 2), 224, 9]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2³², 256, F₂, 8) (dual of [256, 224, 9]-code), using
      - 1 times truncation [i] based on linear OA(2³³, 257, F₂, 9) (dual of [257, 224, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 257Â | 2¹⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(32−8, 32, 561)-Net in Base 2 — Upper bound on s

There is no (24, 32, 562)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 4321 000788 > 2³² [i]