MinT - Best Known (123, 147, s)-Nets in Base 2

Best Known (123, 147, s)-Nets in Base 2

(123, 147, 390)-Net over F₂ — Constructive and digital

Digital (123, 147, 390)-net over F₂, using

2³ times duplication [i] based on digital (120, 144, 390)-net over F₂, using
- trace code for nets [i] based on digital (0, 24, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(123, 147, 1192)-Net over F₂ — Digital

Digital (123, 147, 1192)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁴⁷, 1192, F₂, 3, 24) (dual of [(1192, 3), 3429, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁴⁷, 1370, F₂, 3, 24) (dual of [(1370, 3), 3963, 25]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2¹⁴⁷, 4110, F₂, 24) (dual of [4110, 3963, 25]-code), using
    - 2 times code embedding in larger space [i] based on linear OA(2¹⁴⁵, 4108, F₂, 24) (dual of [4108, 3963, 25]-code), using
      - 1 times truncation [i] based on linear OA(2¹⁴⁶, 4109, F₂, 25) (dual of [4109, 3963, 26]-code), using
        constructionÂ X applied to C_e(24) âŠ‚ C_e(22) [i] based on
        linear OA(2¹⁴⁵, 4096, F₂, 25) (dual of [4096, 3951, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        linear OA(2¹³³, 4096, F₂, 23) (dual of [4096, 3963, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(2¹, 13, F₂, 1) (dual of [13, 12, 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, 147, 25744)-Net in Base 2 — Upper bound on s

There is no (123, 147, 25745)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 178 450599 171125 784452 108298 369654 413305 302684 > 2¹⁴⁷ [i]

Best Known (123, 147, s)-Nets in Base 2

(123, 147, 390)-Net over F2 — Constructive and digital

(123, 147, 1192)-Net over F2 — Digital

(123, 147, 25744)-Net in Base 2 — Upper bound on s

(123, 147, 390)-Net over F₂ — Constructive and digital

(123, 147, 1192)-Net over F₂ — Digital