MinT - Best Known (200−35, 200, s)-Nets in Base 2

Best Known (200−35, 200, s)-Nets in Base 2

(200−35, 200, 380)-Net over F₂ — Constructive and digital

Digital (165, 200, 380)-net over F₂, using

trace code for nets [i] based on digital (5, 40, 76)-net over F₃₂, using
- net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
    - a function field by SÃ©mirat [i]

(200−35, 200, 886)-Net over F₂ — Digital

Digital (165, 200, 886)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰⁰, 886, F₂, 2, 35) (dual of [(886, 2), 1572, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁰⁰, 1046, F₂, 2, 35) (dual of [(1046, 2), 1892, 36]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2²⁰⁰, 2092, F₂, 35) (dual of [2092, 1892, 36]-code), using
    - discarding factors / shortening the dual code based on linear OA(2²⁰⁰, 2093, F₂, 35) (dual of [2093, 1893, 36]-code), using
      - constructionÂ X applied to C_e(34) âŠ‚ C_e(28) [i] based on
        linear OA(2¹⁸⁸, 2048, F₂, 35) (dual of [2048, 1860, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
        linear OA(2¹⁵⁵, 2048, F₂, 29) (dual of [2048, 1893, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
        linear OA(2¹², 45, F₂, 5) (dual of [45, 33, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹², 48, F₂, 5) (dual of [48, 36, 6]-code), using
        adding a parity check bit [i] based on linear OA(2¹¹, 47, F₂, 4) (dual of [47, 36, 5]-code), using
        extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F₂, using
        linear code embedding found in a computer search [i]

(200−35, 200, 23950)-Net in Base 2 — Upper bound on s

There is no (165, 200, 23951)-net in base 2, because

1 times m-reduction [i] would yield (165, 199, 23951)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 803770 843821 675393 432065 600556 660817 128713 483482 068522 567232 > 2¹⁹⁹ [i]

Best Known (200−35, 200, s)-Nets in Base 2

(200−35, 200, 380)-Net over F2 — Constructive and digital

(200−35, 200, 886)-Net over F2 — Digital

(200−35, 200, 23950)-Net in Base 2 — Upper bound on s

(200−35, 200, 380)-Net over F₂ — Constructive and digital

(200−35, 200, 886)-Net over F₂ — Digital