MinT - Best Known (31, 31+10, s)-Nets in Base 2

Best Known (31, 31+10, s)-Nets in Base 2

(31, 31+10, 72)-Net over F₂ — Constructive and digital

Digital (31, 41, 72)-net over F₂, using

1 times m-reduction [i] based on digital (31, 42, 72)-net over F₂, using
- trace code for nets [i] based on digital (3, 14, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]

(31, 31+10, 132)-Net over F₂ — Digital

Digital (31, 41, 132)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁴¹, 132, F₂, 2, 10) (dual of [(132, 2), 223, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁴¹, 264, F₂, 10) (dual of [264, 223, 11]-code), using
  - 1 times truncation [i] based on linear OA(2⁴², 265, F₂, 11) (dual of [265, 223, 12]-code), using
    - constructionÂ X applied to C_e(10) âŠ‚ C_e(8) [i] based on
      1. 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]
      2. linear OA(2³³, 256, F₂, 9) (dual of [256, 223, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [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]

(31, 31+10, 759)-Net in Base 2 — Upper bound on s

There is no (31, 41, 760)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2 211659 892903 > 2⁴¹ [i]