MinT - Best Known (90, 90+20, s)-Nets in Base 2

Best Known (90, 90+20, s)-Nets in Base 2

(90, 90+20, 260)-Net over F₂ — Constructive and digital

Digital (90, 110, 260)-net over F₂, using

2 times m-reduction [i] based on digital (90, 112, 260)-net over F₂, using
- trace code for nets [i] based on digital (6, 28, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(90, 90+20, 678)-Net over F₂ — Digital

Digital (90, 110, 678)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹⁰, 678, F₂, 3, 20) (dual of [(678, 3), 1924, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹¹⁰, 682, F₂, 3, 20) (dual of [(682, 3), 1936, 21]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2¹¹⁰, 2046, F₂, 20) (dual of [2046, 1936, 21]-code), using
    - discarding factors / shortening the dual code based on linear OA(2¹¹⁰, 2048, F₂, 20) (dual of [2048, 1938, 21]-code), using
      - 1 times truncation [i] based on linear OA(2¹¹¹, 2049, F₂, 21) (dual of [2049, 1938, 22]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]

(90, 90+20, 9260)-Net in Base 2 — Upper bound on s

There is no (90, 110, 9261)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1298 989131 158622 120558 271547 129878 > 2¹¹⁰ [i]