Best Known (56−9, 56, s)-Nets in Base 2
(56−9, 56, 2052)-Net over F2 — Constructive and digital
Digital (47, 56, 2052)-net over F2, using
- net defined by OOA [i] based on linear OOA(256, 2052, F2, 9, 9) (dual of [(2052, 9), 18412, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(256, 2052, F2, 8, 9) (dual of [(2052, 8), 16360, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(256, 8209, F2, 9) (dual of [8209, 8153, 10]-code), using
- 2 times code embedding in larger space [i] based on linear OA(254, 8207, F2, 9) (dual of [8207, 8153, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(253, 8192, F2, 9) (dual of [8192, 8139, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(240, 8192, F2, 7) (dual of [8192, 8152, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 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]
- 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
- 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
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(254, 8207, F2, 9) (dual of [8207, 8153, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(256, 8209, F2, 9) (dual of [8209, 8153, 10]-code), using
- appending kth column [i] based on linear OOA(256, 2052, F2, 8, 9) (dual of [(2052, 8), 16360, 10]-NRT-code), using
(56−9, 56, 2736)-Net over F2 — Digital
Digital (47, 56, 2736)-net over F2, using
- 21 times duplication [i] based on 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 Gilbert–Varšamov 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
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(253, 8192, F2, 9) (dual of [8192, 8139, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(240, 8192, F2, 7) (dual of [8192, 8152, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 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]
- 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
- 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
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(254, 8207, F2, 9) (dual of [8207, 8153, 10]-code), using
- OOA 3-folding [i] based on linear OA(255, 8208, F2, 9) (dual of [8208, 8153, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(255, 2736, F2, 3, 9) (dual of [(2736, 3), 8153, 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
- net defined by OOA [i] based on linear OOA(255, 2736, F2, 9, 9) (dual of [(2736, 9), 24569, 10]-NRT-code), using
(56−9, 56, 30488)-Net in Base 2 — Upper bound on s
There is no (47, 56, 30489)-net in base 2, because
- 1 times m-reduction [i] would yield (47, 55, 30489)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 36030 874442 180082 > 255 [i]