MinT - Best Known (145−10, 145, s)-Nets in Base 2

Best Known (145−10, 145, s)-Nets in Base 2

(145−10, 145, 4194306)-Net over F₂ — Constructive and digital

Digital (135, 145, 4194306)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 6, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - nets constructed from net-embeddable BCH codes [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (129, 139, 4194301)-net over F₂, using
  - 1 times m-reduction [i] based on digital (129, 140, 4194301)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2¹⁴⁰, 4194301, F₂, 15, 11) (dual of [(4194301, 15), 62914375, 12]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 11), using
        OOAs constructed from BCH codes [i]

(145−10, 145, large)-Net over F₂ — Digital

Digital (135, 145, large)-net over F₂, using

2⁵ times duplication [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]

(145−10, 145, large)-Net in Base 2 — Upper bound on s

There is no (135, 145, large)-net in base 2, because

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