MinT - Best Known (16, 16+9, s)-Nets in Base 2

Best Known (16, 16+9, s)-Nets in Base 2

(16, 16+9, 34)-Net over F₂ — Constructive and digital

Digital (16, 25, 34)-net over F₂, using

net defined by OOA [i] based on linear OOA(2²⁵, 34, F₂, 9, 9) (dual of [(34, 9), 281, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2²⁵, 34, F₂, 8, 9) (dual of [(34, 8), 247, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁸, 17, F₂, 8, 4) (dual of [(17, 8), 128, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2⁸, 17, F₂, 4, 4) (dual of [(17, 4), 60, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁸, 17, F₂, 3, 4) (dual of [(17, 3), 43, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (4, 8, 17)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
    2. linear OOA(2¹⁷, 17, F₂, 8, 9) (dual of [(17, 8), 119, 10]-NRT-code), using
      - extracting embedded OOA [i] based on digital (8, 17, 17)-net over F₂, using
        a shift-net [i]

(16, 16+9, 136)-Net in Base 2 — Upper bound on s

There is no (16, 25, 137)-net in base 2, because

1 times m-reduction [i] would yield (16, 24, 137)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 17 138838 > 2²⁴ [i]