MinT - Best Known (115−20, 115, s)-Nets in Base 2

Best Known (115−20, 115, s)-Nets in Base 2

(115−20, 115, 320)-Net over F₂ — Constructive and digital

Digital (95, 115, 320)-net over F₂, using

trace code for nets [i] based on digital (3, 23, 64)-net over F₃₂, using
- net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
    - a function field by SÃ©mirat [i]

(115−20, 115, 695)-Net over F₂ — Digital

Digital (95, 115, 695)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹⁵, 695, F₂, 2, 20) (dual of [(695, 2), 1275, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹¹⁵, 1032, F₂, 2, 20) (dual of [(1032, 2), 1949, 21]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2¹¹⁵, 2064, F₂, 20) (dual of [2064, 1949, 21]-code), using
    - 1 times truncation [i] based on linear OA(2¹¹⁶, 2065, F₂, 21) (dual of [2065, 1949, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,8]) [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]
        linear OA(2⁸⁹, 2049, F₂, 17) (dual of [2049, 1960, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(2⁵, 16, F₂, 3) (dual of [16, 11, 4]-code or 16-cap in PG(4,2)), using
        Reedâ€“Muller code RM(1,4) [i]
        caps in base bÂ =Â 2 [i]

(115−20, 115, 13102)-Net in Base 2 — Upper bound on s

There is no (95, 115, 13103)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 41566 678417 513087 036460 999631 263529 > 2¹¹⁵ [i]