MinT - Best Known (51, 69, s)-Nets in Base 2

Best Known (51, 69, s)-Nets in Base 2

(51, 69, 84)-Net over F₂ — Constructive and digital

Digital (51, 69, 84)-net over F₂, using

trace code for nets [i] based on digital (5, 23, 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]

(51, 69, 122)-Net over F₂ — Digital

Digital (51, 69, 122)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁶⁹, 122, F₂, 2, 18) (dual of [(122, 2), 175, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2⁶⁹, 130, F₂, 2, 18) (dual of [(130, 2), 191, 19]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2⁶⁹, 260, F₂, 18) (dual of [260, 191, 19]-code), using
    - 1 times truncation [i] based on linear OA(2⁷⁰, 261, F₂, 19) (dual of [261, 191, 20]-code), using
      - constructionÂ X applied to C_e(18) âŠ‚ C_e(16) [i] based on
        linear OA(2⁶⁹, 256, F₂, 19) (dual of [256, 187, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 255Â = 2⁸âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        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]
        linear OA(2¹, 5, F₂, 1) (dual of [5, 4, 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]

(51, 69, 829)-Net in Base 2 — Upper bound on s

There is no (51, 69, 830)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 591 934945 034753 146027 > 2⁶⁹ [i]