MinT - Best Known (123, 123+28, s)-Nets in Base 2

Best Known (123, 123+28, s)-Nets in Base 2

(123, 123+28, 265)-Net over F₂ — Constructive and digital

Digital (123, 151, 265)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (108, 136, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 34, 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]

(123, 123+28, 542)-Net over F₂ — Digital

Digital (123, 151, 542)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(2¹⁵¹, 542, F₂, 28) (dual of [542, 391, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2¹⁵¹, 1064, F₂, 28) (dual of [1064, 913, 29]-code), using
  - 1 times truncation [i] based on linear OA(2¹⁵², 1065, F₂, 29) (dual of [1065, 913, 30]-code), using
    - constructionÂ XX applied to C₁Â =Â C([1019,22]), C₂Â =Â C([1,24]), C₃Â =Â C₁Â +Â C₂Â =Â C([1,22]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1019,24]) [i] based on
      1. linear OA(2¹³¹, 1023, F₂, 27) (dual of [1023, 892, 28]-code), using the primitive BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â {−4,−3,…,22}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      2. linear OA(2¹²⁰, 1023, F₂, 24) (dual of [1023, 903, 25]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      3. linear OA(2¹⁴¹, 1023, F₂, 29) (dual of [1023, 882, 30]-code), using the primitive BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â {−4,−3,…,24}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
      4. linear OA(2¹¹⁰, 1023, F₂, 22) (dual of [1023, 913, 23]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      5. linear OA(2¹⁰, 31, F₂, 4) (dual of [31, 21, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁰, 32, F₂, 4) (dual of [32, 22, 5]-code), using
        1 times truncation [i] based on linear OA(2¹¹, 33, F₂, 5) (dual of [33, 22, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33Â | 2¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
      6. 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, using
        parity-check code [i]

(123, 123+28, 10652)-Net in Base 2 — Upper bound on s

There is no (123, 151, 10653)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2856 784453 188329 976642 276913 484757 454800 473916 > 2¹⁵¹ [i]

Best Known (123, 123+28, s)-Nets in Base 2

(123, 123+28, 265)-Net over F2 — Constructive and digital

(123, 123+28, 542)-Net over F2 — Digital

(123, 123+28, 10652)-Net in Base 2 — Upper bound on s

(123, 123+28, 265)-Net over F₂ — Constructive and digital

(123, 123+28, 542)-Net over F₂ — Digital