MinT - Best Known (200−40, 200, s)-Nets in Base 2

Best Known (200−40, 200, s)-Nets in Base 2

(200−40, 200, 260)-Net over F₂ — Constructive and digital

Digital (160, 200, 260)-net over F₂, using

t-expansion [i] based on digital (159, 200, 260)-net over F₂, using
- 4 times m-reduction [i] based on digital (159, 204, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 51, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(200−40, 200, 524)-Net over F₂ — Digital

Digital (160, 200, 524)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰⁰, 524, F₂, 2, 40) (dual of [(524, 2), 848, 41]-NRT-code), using
- strength reduction [i] based on linear OOA(2²⁰⁰, 524, F₂, 2, 41) (dual of [(524, 2), 848, 42]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2²⁰⁰, 1048, F₂, 41) (dual of [1048, 848, 42]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(2¹⁹⁹, 1047, F₂, 41) (dual of [1047, 848, 42]-code), using
      - adding a parity check bit [i] based on linear OA(2¹⁹⁸, 1046, F₂, 40) (dual of [1046, 848, 41]-code), using
        constructionÂ XX applied to C₁Â =Â C([1021,36]), C₂Â =Â C([1,38]), C₃Â =Â C₁Â +Â C₂Â =Â C([1,36]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1021,38]) [i] based on
        linear OA(2¹⁸⁶, 1023, F₂, 39) (dual of [1023, 837, 40]-code), using the primitive BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â {−2,−1,…,36}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
        linear OA(2¹⁸⁵, 1023, F₂, 38) (dual of [1023, 838, 39]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
        linear OA(2¹⁹⁶, 1023, F₂, 41) (dual of [1023, 827, 42]-code), using the primitive BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â {−2,−1,…,38}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
        linear OA(2¹⁷⁵, 1023, F₂, 36) (dual of [1023, 848, 37]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [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]
        linear OA(2¹, 11, F₂, 1) (dual of [11, 10, 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 (see above)

(200−40, 200, 8474)-Net in Base 2 — Upper bound on s

There is no (160, 200, 8475)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 609917 261358 773574 450276 042554 824150 631504 726810 995345 561806 > 2²⁰⁰ [i]

Best Known (200−40, 200, s)-Nets in Base 2

(200−40, 200, 260)-Net over F2 — Constructive and digital

(200−40, 200, 524)-Net over F2 — Digital

(200−40, 200, 8474)-Net in Base 2 — Upper bound on s

(200−40, 200, 260)-Net over F₂ — Constructive and digital

(200−40, 200, 524)-Net over F₂ — Digital