MinT - Best Known (122−10, 122, s)-Nets in Base 2

Best Known (122−10, 122, s)-Nets in Base 2

(122−10, 122, 1677727)-Net over F₂ — Constructive and digital

Digital (112, 122, 1677727)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 7)-net over F₂, using
  - a shift-net [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]

(122−10, 122, 2796208)-Net over F₂ — Digital

Digital (112, 122, 2796208)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹²², 2796208, F₂, 3, 10) (dual of [(2796208, 3), 8388502, 11]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁷, 7, F₂, 3, 5) (dual of [(7, 3), 14, 6]-NRT-code), using
    - extracting embedded OOA [i] based on digital (2, 7, 7)-net over F₂, using
      - a shift-net [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]

(122−10, 122, large)-Net in Base 2 — Upper bound on s

There is no (112, 122, large)-net in base 2, because

8 times m-reduction [i] would yield (112, 114, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]