MinT - Best Known (234−43, 234, s)-Nets in Base 2

Best Known (234−43, 234, s)-Nets in Base 2

(234−43, 234, 320)-Net over F₂ — Constructive and digital

Digital (191, 234, 320)-net over F₂, using

1 times m-reduction [i] based on digital (191, 235, 320)-net over F₂, using
- trace code for nets [i] based on digital (3, 47, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]

(234−43, 234, 822)-Net over F₂ — Digital

Digital (191, 234, 822)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³⁴, 822, F₂, 2, 43) (dual of [(822, 2), 1410, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²³⁴, 1030, F₂, 2, 43) (dual of [(1030, 2), 1826, 44]-NRT-code), using
  - 2¹ times duplication [i] based on linear OOA(2²³³, 1030, F₂, 2, 43) (dual of [(1030, 2), 1827, 44]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2²³³, 2060, F₂, 43) (dual of [2060, 1827, 44]-code), using
      - constructionÂ X applied to C_e(42) âŠ‚ C_e(40) [i] based on
        linear OA(2²³², 2048, F₂, 43) (dual of [2048, 1816, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
        linear OA(2²²¹, 2048, F₂, 41) (dual of [2048, 1827, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
        linear OA(2¹, 12, F₂, 1) (dual of [12, 11, 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]

(234−43, 234, 18957)-Net in Base 2 — Upper bound on s

There is no (191, 234, 18958)-net in base 2, because

1 times m-reduction [i] would yield (191, 233, 18958)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13809 851120 306831 215248 526534 610301 128598 144909 255786 932573 296294 167072 > 2²³³ [i]