MinT - Best Known (44, 44+8, s)-Nets in Base 2

Best Known (44, 44+8, s)-Nets in Base 2

(44, 44+8, 2048)-Net over F₂ — Constructive and digital

Digital (44, 52, 2048)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁵², 2048, F₂, 8, 8) (dual of [(2048, 8), 16332, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2⁵², 8192, F₂, 8) (dual of [8192, 8140, 9]-code), using
  - 1 times truncation [i] based on linear OA(2⁵³, 8193, F₂, 9) (dual of [8193, 8140, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(44, 44+8, 2730)-Net over F₂ — Digital

Digital (44, 52, 2730)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁵², 2730, F₂, 8, 8) (dual of [(2730, 8), 21788, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁵², 2730, F₂, 7, 8) (dual of [(2730, 7), 19058, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁵², 2730, F₂, 3, 8) (dual of [(2730, 3), 8138, 9]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2⁵², 8190, F₂, 8) (dual of [8190, 8138, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(2⁵², 8192, F₂, 8) (dual of [8192, 8140, 9]-code), using
        1 times truncation [i] based on linear OA(2⁵³, 8193, F₂, 9) (dual of [8193, 8140, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8193Â | 2²⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(44, 44+8, 18126)-Net in Base 2 — Upper bound on s

There is no (44, 52, 18127)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 4504 218355 950673 > 2⁵² [i]