MinT - Best Known (240−41, 240, s)-Nets in Base 2

Best Known (240−41, 240, s)-Nets in Base 2

(240−41, 240, 490)-Net over F₂ — Constructive and digital

Digital (199, 240, 490)-net over F₂, using

trace code for nets [i] based on digital (7, 48, 98)-net over F₃₂, using
- net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
    - a function field by SÃ©mirat [i]

(240−41, 240, 1056)-Net over F₂ — Digital

Digital (199, 240, 1056)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴⁰, 1056, F₂, 2, 41) (dual of [(1056, 2), 1872, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2²⁴⁰, 2112, F₂, 41) (dual of [2112, 1872, 42]-code), using
  - constructionÂ X applied to C([0,20]) âŠ‚ C([0,16]) [i] based on
    1. linear OA(2²²¹, 2049, F₂, 41) (dual of [2049, 1828, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
    2. linear OA(2¹⁷⁷, 2049, F₂, 33) (dual of [2049, 1872, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    3. linear OA(2¹⁹, 63, F₂, 7) (dual of [63, 44, 8]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 2⁶âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(240−41, 240, 32826)-Net in Base 2 — Upper bound on s

There is no (199, 240, 32827)-net in base 2, because

1 times m-reduction [i] would yield (199, 239, 32827)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 883686 398932 550212 703576 073311 148143 486305 502636 131608 081450 909130 342666 > 2²³⁹ [i]