MinT - Best Known (202, 241, s)-Nets in Base 2

Best Known (202, 241, s)-Nets in Base 2

(202, 241, 520)-Net over F₂ — Constructive and digital

Digital (202, 241, 520)-net over F₂, using

2¹ times duplication [i] based on digital (201, 240, 520)-net over F₂, using
- trace code for nets [i] based on digital (9, 48, 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]

(202, 241, 1381)-Net over F₂ — Digital

Digital (202, 241, 1381)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴¹, 1381, F₂, 3, 39) (dual of [(1381, 3), 3902, 40]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2²⁴¹, 4143, F₂, 39) (dual of [4143, 3902, 40]-code), using
  - discarding factors / shortening the dual code based on linear OA(2²⁴¹, 4144, F₂, 39) (dual of [4144, 3903, 40]-code), using
    - constructionÂ X applied to C_e(38) âŠ‚ C_e(32) [i] based on
      1. linear OA(2²²⁹, 4096, F₂, 39) (dual of [4096, 3867, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
      2. linear OA(2¹⁹³, 4096, F₂, 33) (dual of [4096, 3903, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
      3. 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]

(202, 241, 50287)-Net in Base 2 — Upper bound on s

There is no (202, 241, 50288)-net in base 2, because

1 times m-reduction [i] would yield (202, 240, 50288)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 767340 688787 771561 178604 592477 610498 821240 152201 398249 298674 635909 501822 > 2²⁴⁰ [i]