MinT - Best Known (240−37, 240, s)-Nets in Base 2

Best Known (240−37, 240, s)-Nets in Base 2

(240−37, 240, 624)-Net over F₂ — Constructive and digital

Digital (203, 240, 624)-net over F₂, using

trace code for nets [i] based on digital (3, 40, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(240−37, 240, 2052)-Net over F₂ — Digital

Digital (203, 240, 2052)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴⁰, 2052, F₂, 4, 37) (dual of [(2052, 4), 7968, 38]-NRT-code), using
- 2² times duplication [i] based on linear OOA(2²³⁸, 2052, F₂, 4, 37) (dual of [(2052, 4), 7970, 38]-NRT-code), using
  - OOA 4-folding [i] based on linear OA(2²³⁸, 8208, F₂, 37) (dual of [8208, 7970, 38]-code), using
    - 2 times code embedding in larger space [i] based on linear OA(2²³⁶, 8206, F₂, 37) (dual of [8206, 7970, 38]-code), using
      - constructionÂ X applied to C_e(36) âŠ‚ C_e(34) [i] based on
        linear OA(2²³⁵, 8192, F₂, 37) (dual of [8192, 7957, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
        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]
        linear OA(2¹, 14, F₂, 1) (dual of [14, 13, 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]

(240−37, 240, 74987)-Net in Base 2 — Upper bound on s

There is no (203, 240, 74988)-net in base 2, because

1 times m-reduction [i] would yield (203, 239, 74988)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 883634 117465 801715 872914 117361 269897 722172 321678 319995 364720 928546 021874 > 2²³⁹ [i]