MinT - Best Known (166−12, 166, s)-Nets in Base 2

Best Known (166−12, 166, s)-Nets in Base 2

(166−12, 166, 1398610)-Net over F₂ — Constructive and digital

Digital (154, 166, 1398610)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (22, 28, 510)-net over F₂, using
  - 1 times m-reduction [i] based on digital (22, 29, 510)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2²⁹, 510, F₂, 7, 7) (dual of [(510, 7), 3541, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(2²⁹, 511, F₂, 3, 7) (dual of [(511, 3), 1504, 8]-NRT-code), using
        OOAs constructed from BCH codes [i]
2. digital (126, 138, 1398100)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹³⁸, 1398100, F₂, 12, 12) (dual of [(1398100, 12), 16777062, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(2¹³⁸, 8388600, F₂, 12) (dual of [8388600, 8388462, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹³⁸, large, F₂, 12) (dual of [large, large−138, 13]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(166−12, 166, 2796712)-Net over F₂ — Digital

Digital (154, 166, 2796712)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶⁶, 2796712, F₂, 3, 12) (dual of [(2796712, 3), 8389970, 13]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²⁸, 511, F₂, 3, 6) (dual of [(511, 3), 1505, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁸, 511, F₂, 2, 6) (dual of [(511, 2), 994, 7]-NRT-code), using
      - OOAs constructed from BCH codes [i]
  2. linear OOA(2¹³⁸, 2796201, F₂, 3, 12) (dual of [(2796201, 3), 8388465, 13]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2¹³⁸, large, F₂, 12) (dual of [large, large−138, 13]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(166−12, 166, large)-Net in Base 2 — Upper bound on s

There is no (154, 166, large)-net in base 2, because

10 times m-reduction [i] would yield (154, 156, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]