MinT - Best Known (58−15, 58, s)-Nets in Base 2

Best Known (58−15, 58, s)-Nets in Base 2

(58−15, 58, 82)-Net over F₂ — Constructive and digital

Digital (43, 58, 82)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 18, 41)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (25, 40, 41)-net over F₂, using
  - 1 times m-reduction [i] based on digital (25, 41, 41)-net over F₂, using
    - a shift-net [i]

(58−15, 58, 119)-Net over F₂ — Digital

Digital (43, 58, 119)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁵⁸, 119, F₂, 2, 15) (dual of [(119, 2), 180, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2⁵⁸, 132, F₂, 2, 15) (dual of [(132, 2), 206, 16]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2⁵⁸, 264, F₂, 15) (dual of [264, 206, 16]-code), using
    - discarding factors / shortening the dual code based on linear OA(2⁵⁸, 265, F₂, 15) (dual of [265, 207, 16]-code), using
      - constructionÂ X applied to C_e(14) âŠ‚ C_e(12) [i] based on
        linear OA(2⁵⁷, 256, F₂, 15) (dual of [256, 199, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(2⁴⁹, 256, F₂, 13) (dual of [256, 207, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(2¹, 9, F₂, 1) (dual of [9, 8, 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]

(58−15, 58, 945)-Net in Base 2 — Upper bound on s

There is no (43, 58, 946)-net in base 2, because

1 times m-reduction [i] would yield (43, 57, 946)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 144753 497280 290504 > 2⁵⁷ [i]