MinT - Best Known (238−37, 238, s)-Nets in Base 2

Best Known (238−37, 238, s)-Nets in Base 2

(238−37, 238, 520)-Net over F₂ — Constructive and digital

Digital (201, 238, 520)-net over F₂, using

2 times m-reduction [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]

(238−37, 238, 1987)-Net over F₂ — Digital

Digital (201, 238, 1987)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁸, 1987, F₂, 4, 37) (dual of [(1987, 4), 7710, 38]-NRT-code), using
- discarding factors / shortening the dual code 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]

(238−37, 238, 69426)-Net in Base 2 — Upper bound on s

There is no (201, 238, 69427)-net in base 2, because

1 times m-reduction [i] would yield (201, 237, 69427)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 220878 230275 756121 890952 788285 816338 104993 444375 468163 119019 666374 856515 > 2²³⁷ [i]