MinT - Best Known (179, 179+47, s)-Nets in Base 2

Best Known (179, 179+47, s)-Nets in Base 2

(179, 179+47, 260)-Net over F₂ — Constructive and digital

Digital (179, 226, 260)-net over F₂, using

t-expansion [i] based on digital (177, 226, 260)-net over F₂, using
- 2 times m-reduction [i] based on digital (177, 228, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 57, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(179, 179+47, 512)-Net over F₂ — Digital

Digital (179, 226, 512)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²²⁶, 512, F₂, 2, 47) (dual of [(512, 2), 798, 48]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2²²⁶, 1024, F₂, 47) (dual of [1024, 798, 48]-code), using
  - an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]

(179, 179+47, 8270)-Net in Base 2 — Upper bound on s

There is no (179, 226, 8271)-net in base 2, because

1 times m-reduction [i] would yield (179, 225, 8271)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 53 981001 242598 023719 843047 109819 012183 883035 595353 612966 207174 208008 > 2²²⁵ [i]