MinT - Best Known (113, 123, s)-Nets in Base 2

Best Known (113, 123, s)-Nets in Base 2

(113, 123, 1677730)-Net over F₂ — Constructive and digital

Digital (113, 123, 1677730)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 8, 10)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (105, 115, 1677720)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹¹⁵, 1677720, F₂, 10, 10) (dual of [(1677720, 10), 16777085, 11]-NRT-code), using
    - OA 5-folding and stacking [i] based on linear OA(2¹¹⁵, 8388600, F₂, 10) (dual of [8388600, 8388485, 11]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹¹⁵, large, F₂, 10) (dual of [large, large−115, 11]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(113, 123, 2796211)-Net over F₂ — Digital

Digital (113, 123, 2796211)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹²³, 2796211, F₂, 3, 10) (dual of [(2796211, 3), 8388510, 11]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁸, 10, F₂, 3, 5) (dual of [(10, 3), 22, 6]-NRT-code), using
    - extracting embedded OOA [i] based on digital (3, 8, 10)-net over F₂, using
      - linear code embedding found in a computer search [i]
  2. linear OOA(2¹¹⁵, 2796201, F₂, 3, 10) (dual of [(2796201, 3), 8388488, 11]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2¹¹⁵, large, F₂, 10) (dual of [large, large−115, 11]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(113, 123, large)-Net in Base 2 — Upper bound on s

There is no (113, 123, large)-net in base 2, because

8 times m-reduction [i] would yield (113, 115, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]