MinT - Best Known (135, 135+8, s)-Nets in Base 2

Best Known (135, 135+8, s)-Nets in Base 2

(135, 135+8, 4210686)-Net over F₂ — Constructive and digital

Digital (135, 143, 4210686)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁴³, 4210686, F₂, 8, 8) (dual of [(4210686, 8), 33685345, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁴³, 4210686, F₂, 7, 8) (dual of [(4210686, 7), 29474659, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2²⁸, 16385, F₂, 7, 4) (dual of [(16385, 7), 114667, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2²⁸, 16385, F₂, 4, 4) (dual of [(16385, 4), 65512, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2²⁸, 16385, F₂, 3, 4) (dual of [(16385, 3), 49127, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (24, 28, 16385)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
    2. linear OOA(2¹¹⁵, 4194301, F₂, 7, 8) (dual of [(4194301, 7), 29359992, 9]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(2¹¹⁵, large, F₂, 2, 8), using
        OOAs constructed from BCH codes [i]

(135, 135+8, large)-Net over F₂ — Digital

Digital (135, 143, large)-net over F₂, using

2³ times duplication [i] based on digital (132, 140, large)-net over F₂, using
- t-expansion [i] based on digital (130, 140, large)-net over F₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 10), using
    - strength reduction [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 11), using
      - OOAs constructed from BCH codes [i]

(135, 135+8, large)-Net in Base 2 — Upper bound on s

There is no (135, 143, large)-net in base 2, because

6 times m-reduction [i] would yield (135, 137, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]