MinT - Best Known (49−12, 49, s)-Nets in Base 2

Best Known (49−12, 49, s)-Nets in Base 2

(49−12, 49, 75)-Net over F₂ — Constructive and digital

Digital (37, 49, 75)-net over F₂, using

2¹ times duplication [i] based on digital (36, 48, 75)-net over F₂, using
- trace code for nets [i] based on digital (4, 16, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]

(49−12, 49, 132)-Net over F₂ — Digital

Digital (37, 49, 132)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁴⁹, 132, F₂, 2, 12) (dual of [(132, 2), 215, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁴⁹, 264, F₂, 12) (dual of [264, 215, 13]-code), using
  - 1 times truncation [i] based on linear OA(2⁵⁰, 265, F₂, 13) (dual of [265, 215, 14]-code), using
    - constructionÂ X applied to C_e(12) âŠ‚ C_e(10) [i] based on
      1. linear OA(2⁴⁹, 256, F₂, 13) (dual of [256, 207, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      2. linear OA(2⁴¹, 256, F₂, 11) (dual of [256, 215, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(2¹, 9, F₂, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(49−12, 49, 851)-Net in Base 2 — Upper bound on s

There is no (37, 49, 852)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 563 718810 047516 > 2⁴⁹ [i]