MinT - Best Known (233, 254, s)-Nets in Base 2

Best Known (233, 254, s)-Nets in Base 2

(233, 254, 838885)-Net over F₂ — Constructive and digital

Digital (233, 254, 838885)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (13, 23, 25)-net over F₂, using
  - 2 times m-reduction [i] based on digital (13, 25, 25)-net over F₂, using
    - a shift-net [i]
2. digital (210, 231, 838860)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²³¹, 838860, F₂, 21, 21) (dual of [(838860, 21), 17615829, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(2²³¹, 8388601, F₂, 21) (dual of [8388601, 8388370, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(2²³¹, large, F₂, 21) (dual of [large, large−231, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(233, 254, 1300117)-Net over F₂ — Digital

Digital (233, 254, 1300117)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁵⁴, 1300117, F₂, 6, 21) (dual of [(1300117, 6), 7800448, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁵⁴, 1398113, F₂, 6, 21) (dual of [(1398113, 6), 8388424, 22]-NRT-code), using
  - OOA 2-folding [i] based on linear OOA(2²⁵⁴, 2796226, F₂, 3, 21) (dual of [(2796226, 3), 8388424, 22]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2²³, 25, F₂, 3, 10) (dual of [(25, 3), 52, 11]-NRT-code), using
        extracting embedded OOA [i] based on digital (13, 23, 25)-net over F₂, using
        2 times m-reduction [i] based on digital (13, 25, 25)-net over F₂, using
        a shift-net [i]
      2. linear OOA(2²³¹, 2796201, F₂, 3, 21) (dual of [(2796201, 3), 8388372, 22]-NRT-code), using
        OOA 3-folding [i] based on linear OA(2²³¹, large, F₂, 21) (dual of [large, large−231, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(233, 254, large)-Net in Base 2 — Upper bound on s

There is no (233, 254, large)-net in base 2, because

19 times m-reduction [i] would yield (233, 235, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]