MinT - Best Known (163, 200, s)-Nets in Base 2

Best Known (163, 200, s)-Nets in Base 2

(163, 200, 320)-Net over F₂ — Constructive and digital

Digital (163, 200, 320)-net over F₂, using

trace code for nets [i] based on digital (3, 40, 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]

(163, 200, 718)-Net over F₂ — Digital

Digital (163, 200, 718)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰⁰, 718, F₂, 2, 37) (dual of [(718, 2), 1236, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁰⁰, 1030, F₂, 2, 37) (dual of [(1030, 2), 1860, 38]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2²⁰⁰, 2060, F₂, 37) (dual of [2060, 1860, 38]-code), using
    - constructionÂ X applied to C_e(36) âŠ‚ C_e(34) [i] based on
      1. linear OA(2¹⁹⁹, 2048, F₂, 37) (dual of [2048, 1849, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
      2. 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]
      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]

(163, 200, 16049)-Net in Base 2 — Upper bound on s

There is no (163, 200, 16050)-net in base 2, because

1 times m-reduction [i] would yield (163, 199, 16050)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 803704 525538 407168 566272 410974 775539 108463 547654 151004 245416 > 2¹⁹⁹ [i]