MinT - Best Known (55−9, 55, s)-Nets in Base 2

Best Known (55−9, 55, s)-Nets in Base 2

(55−9, 55, 2051)-Net over F₂ — Constructive and digital

Digital (46, 55, 2051)-net over F₂, using

2¹ times duplication [i] based on digital (45, 54, 2051)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2⁵⁴, 2051, F₂, 9, 9) (dual of [(2051, 9), 18405, 10]-NRT-code), using
  - appending kth column [i] based on linear OOA(2⁵⁴, 2051, F₂, 8, 9) (dual of [(2051, 8), 16354, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(2⁵⁴, 8205, F₂, 9) (dual of [8205, 8151, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(2⁵⁴, 8206, F₂, 9) (dual of [8206, 8152, 10]-code), using
        constructionÂ X applied to C_e(8) âŠ‚ C_e(6) [i] based on
        linear OA(2⁵³, 8192, F₂, 9) (dual of [8192, 8139, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(2⁴⁰, 8192, F₂, 7) (dual of [8192, 8152, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(2¹, 14, F₂, 1) (dual of [14, 13, 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]

(55−9, 55, 2736)-Net over F₂ — Digital

Digital (46, 55, 2736)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁵⁵, 2736, F₂, 9, 9) (dual of [(2736, 9), 24569, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁵⁵, 2736, F₂, 8, 9) (dual of [(2736, 8), 21833, 10]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁵⁵, 2736, F₂, 3, 9) (dual of [(2736, 3), 8153, 10]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2⁵⁵, 8208, F₂, 9) (dual of [8208, 8153, 10]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(2⁵⁴, 8207, F₂, 9) (dual of [8207, 8153, 10]-code), using
        constructionÂ X4 applied to C_e(8) âŠ‚ C_e(6) [i] based on
        linear OA(2⁵³, 8192, F₂, 9) (dual of [8192, 8139, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(2⁴⁰, 8192, F₂, 7) (dual of [8192, 8152, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(2¹⁴, 15, F₂, 13) (dual of [15, 1, 14]-code), using
        strength reduction [i] based on linear OA(2¹⁴, 15, F₂, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
        dual of repetition code with length 15 [i]
        linear OA(2¹, 15, F₂, 1) (dual of [15, 14, 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]

(55−9, 55, 25636)-Net in Base 2 — Upper bound on s

There is no (46, 55, 25637)-net in base 2, because

1 times m-reduction [i] would yield (46, 54, 25637)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 18014 830560 202713 > 2⁵⁴ [i]

Best Known (55−9, 55, s)-Nets in Base 2

(55−9, 55, 2051)-Net over F2 — Constructive and digital

(55−9, 55, 2736)-Net over F2 — Digital

(55−9, 55, 25636)-Net in Base 2 — Upper bound on s

(55−9, 55, 2051)-Net over F₂ — Constructive and digital

(55−9, 55, 2736)-Net over F₂ — Digital