Best Known (58−9, 58, s)-Nets in Base 2
(58−9, 58, 4099)-Net over F2 — Constructive and digital
Digital (49, 58, 4099)-net over F2, using
- net defined by OOA [i] based on linear OOA(258, 4099, F2, 9, 9) (dual of [(4099, 9), 36833, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(258, 4099, F2, 8, 9) (dual of [(4099, 8), 32734, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(258, 16397, F2, 9) (dual of [16397, 16339, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(258, 16399, F2, 9) (dual of [16399, 16341, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(257, 16384, F2, 9) (dual of [16384, 16327, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(243, 16384, F2, 7) (dual of [16384, 16341, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(258, 16399, F2, 9) (dual of [16399, 16341, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(258, 16397, F2, 9) (dual of [16397, 16339, 10]-code), using
- appending kth column [i] based on linear OOA(258, 4099, F2, 8, 9) (dual of [(4099, 8), 32734, 10]-NRT-code), using
(58−9, 58, 5329)-Net over F2 — Digital
Digital (49, 58, 5329)-net over F2, using
- net defined by OOA [i] based on linear OOA(258, 5329, F2, 9, 9) (dual of [(5329, 9), 47903, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(258, 5329, F2, 8, 9) (dual of [(5329, 8), 42574, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(258, 5329, F2, 3, 9) (dual of [(5329, 3), 15929, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(258, 5466, F2, 3, 9) (dual of [(5466, 3), 16340, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(258, 16398, F2, 9) (dual of [16398, 16340, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(258, 16399, F2, 9) (dual of [16399, 16341, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(257, 16384, F2, 9) (dual of [16384, 16327, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(243, 16384, F2, 7) (dual of [16384, 16341, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(258, 16399, F2, 9) (dual of [16399, 16341, 10]-code), using
- OOA 3-folding [i] based on linear OA(258, 16398, F2, 9) (dual of [16398, 16340, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(258, 5466, F2, 3, 9) (dual of [(5466, 3), 16340, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(258, 5329, F2, 3, 9) (dual of [(5329, 3), 15929, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(258, 5329, F2, 8, 9) (dual of [(5329, 8), 42574, 10]-NRT-code), using
(58−9, 58, 43119)-Net in Base 2 — Upper bound on s
There is no (49, 58, 43120)-net in base 2, because
- 1 times m-reduction [i] would yield (49, 57, 43120)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 144120 353899 594381 > 257 [i]