MinT - Best Known (36−8, 36, s)-Nets in Base 2

Best Known (36−8, 36, s)-Nets in Base 2

(36−8, 36, 128)-Net over F₂ — Constructive and digital

Digital (28, 36, 128)-net over F₂, using

net defined by OOA [i] based on linear OOA(2³⁶, 128, F₂, 8, 8) (dual of [(128, 8), 988, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2³⁶, 512, F₂, 8) (dual of [512, 476, 9]-code), using
  - 1 times truncation [i] based on linear OA(2³⁷, 513, F₂, 9) (dual of [513, 476, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 513Â | 2¹⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(36−8, 36, 256)-Net over F₂ — Digital

Digital (28, 36, 256)-net over F₂, using

net defined by OOA [i] based on linear OOA(2³⁶, 256, F₂, 8, 8) (dual of [(256, 8), 2012, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2³⁶, 256, F₂, 7, 8) (dual of [(256, 7), 1756, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³⁶, 256, F₂, 2, 8) (dual of [(256, 2), 476, 9]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2³⁶, 512, F₂, 8) (dual of [512, 476, 9]-code), using
      - 1 times truncation [i] based on linear OA(2³⁷, 513, F₂, 9) (dual of [513, 476, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 513Â | 2¹⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(36−8, 36, 1127)-Net in Base 2 — Upper bound on s

There is no (28, 36, 1128)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 68779 309979 > 2³⁶ [i]