MinT - Best Known (65−16, 65, s)-Nets in Base 2

Best Known (65−16, 65, s)-Nets in Base 2

(65−16, 65, 84)-Net over F₂ — Constructive and digital

Digital (49, 65, 84)-net over F₂, using

1 times m-reduction [i] based on digital (49, 66, 84)-net over F₂, using
- trace code for nets [i] based on digital (5, 22, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]

(65−16, 65, 132)-Net over F₂ — Digital

Digital (49, 65, 132)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁶⁵, 132, F₂, 2, 16) (dual of [(132, 2), 199, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁶⁵, 264, F₂, 16) (dual of [264, 199, 17]-code), using
  - 1 times truncation [i] based on linear OA(2⁶⁶, 265, F₂, 17) (dual of [265, 199, 18]-code), using
    - constructionÂ X applied to C_e(16) âŠ‚ C_e(14) [i] based on
      1. linear OA(2⁶⁵, 256, F₂, 17) (dual of [256, 191, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
      2. 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]
      3. 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]

(65−16, 65, 1039)-Net in Base 2 — Upper bound on s

There is no (49, 65, 1040)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 37 048484 975820 270499 > 2⁶⁵ [i]