MinT - Best Known (221−34, 221, s)-Nets in Base 2

Best Known (221−34, 221, s)-Nets in Base 2

(221−34, 221, 520)-Net over F₂ — Constructive and digital

Digital (187, 221, 520)-net over F₂, using

2¹ times duplication [i] based on digital (186, 220, 520)-net over F₂, using
- t-expansion [i] based on digital (185, 220, 520)-net over F₂, using
  - trace code for nets [i] based on digital (9, 44, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(221−34, 221, 2046)-Net over F₂ — Digital

Digital (187, 221, 2046)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²²¹, 2046, F₂, 4, 34) (dual of [(2046, 4), 7963, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²²¹, 2047, F₂, 4, 34) (dual of [(2047, 4), 7967, 35]-NRT-code), using
  - OOA 4-folding [i] based on linear OA(2²²¹, 8188, F₂, 34) (dual of [8188, 7967, 35]-code), using
    - discarding factors / shortening the dual code based on linear OA(2²²¹, 8191, F₂, 34) (dual of [8191, 7970, 35]-code), using
      - 1 times truncation [i] based on linear OA(2²²², 8192, F₂, 35) (dual of [8192, 7970, 36]-code), using
        an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(221−34, 221, 58769)-Net in Base 2 — Upper bound on s

There is no (187, 221, 58770)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 370680 498535 854746 366259 460725 528113 637081 816276 825899 428831 498203 > 2²²¹ [i]