MinT - Best Known (152−28, 152, s)-Nets in Base 2

Best Known (152−28, 152, s)-Nets in Base 2

(152−28, 152, 266)-Net over F₂ — Constructive and digital

Digital (124, 152, 266)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 16, 6)-net over F₂, using
  - net from sequence [i] based on digital (2, 5)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 2 and N(F) ≥ 6, 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]

(152−28, 152, 557)-Net over F₂ — Digital

Digital (124, 152, 557)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(2¹⁵², 557, F₂, 28) (dual of [557, 405, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2¹⁵², 1062, F₂, 28) (dual of [1062, 910, 29]-code), using
  - constructionÂ X applied to C([1,28]) âŠ‚ C([1,22]) [i] based on
    1. linear OA(2¹⁴⁰, 1023, F₂, 28) (dual of [1023, 883, 29]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. 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]
    3. linear OA(2¹², 39, F₂, 5) (dual of [39, 27, 6]-code), using
      - constructionÂ X4 applied to C_e(4) âŠ‚ C_e(2) [i] based on
        linear OA(2¹¹, 32, F₂, 5) (dual of [32, 21, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 31Â = 2⁵âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(2⁶, 32, F₂, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 31Â = 2⁵âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(2⁶, 7, F₂, 5) (dual of [7, 1, 6]-code), using
        strength reduction [i] based on linear OA(2⁶, 7, F₂, 6) (dual of [7, 1, 7]-code or 7-arc in PG(5,2)), using
        dual of repetition code with length 7 [i]
        linear OA(2¹, 7, F₂, 1) (dual of [7, 6, 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]

(152−28, 152, 11194)-Net in Base 2 — Upper bound on s

There is no (124, 152, 11195)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5715 357791 269759 856534 485564 120057 169907 811164 > 2¹⁵² [i]

Best Known (152−28, 152, s)-Nets in Base 2

(152−28, 152, 266)-Net over F2 — Constructive and digital

(152−28, 152, 557)-Net over F2 — Digital

(152−28, 152, 11194)-Net in Base 2 — Upper bound on s

(152−28, 152, 266)-Net over F₂ — Constructive and digital

(152−28, 152, 557)-Net over F₂ — Digital