MinT - Best Known (43, 56, s)-Nets in Base 2

Best Known (43, 56, s)-Nets in Base 2

(43, 56, 96)-Net over F₂ — Constructive and digital

Digital (43, 56, 96)-net over F₂, using

trace code for nets [i] based on digital (1, 14, 24)-net over F₁₆, using
- net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(43, 56, 178)-Net over F₂ — Digital

Digital (43, 56, 178)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁵⁶, 178, F₂, 2, 13) (dual of [(178, 2), 300, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2⁵⁶, 261, F₂, 2, 13) (dual of [(261, 2), 466, 14]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2⁵⁶, 522, F₂, 13) (dual of [522, 466, 14]-code), using
    - constructionÂ X applied to C_e(12) âŠ‚ C_e(10) [i] based on
      1. linear OA(2⁵⁵, 512, F₂, 13) (dual of [512, 457, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 511Â = 2⁹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      2. linear OA(2⁴⁶, 512, F₂, 11) (dual of [512, 466, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 511Â = 2⁹âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      3. linear OA(2¹, 10, F₂, 1) (dual of [10, 9, 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]

(43, 56, 1712)-Net in Base 2 — Upper bound on s

There is no (43, 56, 1713)-net in base 2, because

1 times m-reduction [i] would yield (43, 55, 1713)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 36147 816264 505096 > 2⁵⁵ [i]