MinT - Best Known (150−25, 150, s)-Nets in Base 2

Best Known (150−25, 150, s)-Nets in Base 2

(150−25, 150, 390)-Net over F₂ — Constructive and digital

Digital (125, 150, 390)-net over F₂, using

trace code for nets [i] based on digital (0, 25, 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]

(150−25, 150, 1081)-Net over F₂ — Digital

Digital (125, 150, 1081)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁵⁰, 1081, F₂, 3, 25) (dual of [(1081, 3), 3093, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁵⁰, 1371, F₂, 3, 25) (dual of [(1371, 3), 3963, 26]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(2¹⁴⁷, 1370, F₂, 3, 25) (dual of [(1370, 3), 3963, 26]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2¹⁴⁷, 4110, F₂, 25) (dual of [4110, 3963, 26]-code), using
      - 1 times code embedding in larger space [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]

(150−25, 150, 28899)-Net in Base 2 — Upper bound on s

There is no (125, 150, 28900)-net in base 2, because

1 times m-reduction [i] would yield (125, 149, 28900)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 713 823937 315412 642847 006468 272390 246407 521576 > 2¹⁴⁹ [i]

Best Known (150−25, 150, s)-Nets in Base 2

(150−25, 150, 390)-Net over F2 — Constructive and digital

(150−25, 150, 1081)-Net over F2 — Digital

(150−25, 150, 28899)-Net in Base 2 — Upper bound on s

(150−25, 150, 390)-Net over F₂ — Constructive and digital

(150−25, 150, 1081)-Net over F₂ — Digital