MinT - Best Known (167−12, 167, s)-Nets in Base 2

Best Known (167−12, 167, s)-Nets in Base 2

(167−12, 167, 1398610)-Net over F₂ — Constructive and digital

Digital (155, 167, 1398610)-net over F₂, using

t-expansion [i] based on digital (154, 167, 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, 139, 1398100)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2¹³⁹, 1398100, F₂, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
      - OOA 6-folding and stacking with additional row [i] based on linear OA(2¹³⁹, 8388601, F₂, 13) (dual of [8388601, 8388462, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(167−12, 167, 2796712)-Net over F₂ — Digital

Digital (155, 167, 2796712)-net over F₂, using

2¹ times duplication [i] based on 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]

(167−12, 167, large)-Net in Base 2 — Upper bound on s

There is no (155, 167, large)-net in base 2, because

10 times m-reduction [i] would yield (155, 157, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]