MinT - Best Known (255−47, 255, s)-Nets in Base 2

Best Known (255−47, 255, s)-Nets in Base 2

(255−47, 255, 320)-Net over F₂ — Constructive and digital

Digital (208, 255, 320)-net over F₂, using

t-expansion [i] based on digital (207, 255, 320)-net over F₂, using
- trace code for nets [i] based on digital (3, 51, 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]

(255−47, 255, 866)-Net over F₂ — Digital

Digital (208, 255, 866)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁵⁵, 866, F₂, 2, 47) (dual of [(866, 2), 1477, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁵⁵, 1030, F₂, 2, 47) (dual of [(1030, 2), 1805, 48]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2²⁵⁵, 2060, F₂, 47) (dual of [2060, 1805, 48]-code), using
    - constructionÂ X applied to C_e(46) âŠ‚ C_e(44) [i] based on
      1. linear OA(2²⁵⁴, 2048, F₂, 47) (dual of [2048, 1794, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
      2. linear OA(2²⁴³, 2048, F₂, 45) (dual of [2048, 1805, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
      3. linear OA(2¹, 12, F₂, 1) (dual of [12, 11, 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]

(255−47, 255, 19867)-Net in Base 2 — Upper bound on s

There is no (208, 255, 19868)-net in base 2, because

1 times m-reduction [i] would yield (208, 254, 19868)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 28981 233994 029505 846333 528344 496381 097992 457210 367014 131481 748321 913711 366386 > 2²⁵⁴ [i]