MinT - Best Known (253−21, 253, s)-Nets in Base 2

Best Known (253−21, 253, s)-Nets in Base 2

(253−21, 253, 838883)-Net over F₂ — Constructive and digital

Digital (232, 253, 838883)-net over F₂, using

2¹ times duplication [i] based on digital (231, 252, 838883)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 21, 23)-net over F₂, using
    - linear code embedding found in a computer search [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]

(253−21, 253, 1237314)-Net over F₂ — Digital

Digital (232, 253, 1237314)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁵³, 1237314, F₂, 6, 21) (dual of [(1237314, 6), 7423631, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁵³, 1398123, F₂, 6, 21) (dual of [(1398123, 6), 8388485, 22]-NRT-code), using
  - 2¹ times duplication [i] based on linear OOA(2²⁵², 1398123, F₂, 6, 21) (dual of [(1398123, 6), 8388486, 22]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2²¹, 23, F₂, 6, 10) (dual of [(23, 6), 117, 11]-NRT-code), using
        extracting embedded OOA [i] based on digital (11, 21, 23)-net over F₂, using
        linear code embedding found in a computer search [i]
      2. linear OOA(2²³¹, 1398100, F₂, 6, 21) (dual of [(1398100, 6), 8388369, 22]-NRT-code), using
        OOA 6-folding [i] based on linear OA(2²³¹, 8388600, F₂, 21) (dual of [8388600, 8388369, 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]

(253−21, 253, large)-Net in Base 2 — Upper bound on s

There is no (232, 253, large)-net in base 2, because

19 times m-reduction [i] would yield (232, 234, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]