Best Known (171−24, 171, s)-Nets in Base 2
(171−24, 171, 1366)-Net over F2 — Constructive and digital
Digital (147, 171, 1366)-net over F2, using
- 21 times duplication [i] based on digital (146, 170, 1366)-net over F2, using
- t-expansion [i] based on digital (145, 170, 1366)-net over F2, using
- net defined by OOA [i] based on linear OOA(2170, 1366, F2, 25, 25) (dual of [(1366, 25), 33980, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2170, 16393, F2, 25) (dual of [16393, 16223, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2170, 16399, F2, 25) (dual of [16399, 16229, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2169, 16384, F2, 25) (dual of [16384, 16215, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2170, 16399, F2, 25) (dual of [16399, 16229, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2170, 16393, F2, 25) (dual of [16393, 16223, 26]-code), using
- net defined by OOA [i] based on linear OOA(2170, 1366, F2, 25, 25) (dual of [(1366, 25), 33980, 26]-NRT-code), using
- t-expansion [i] based on digital (145, 170, 1366)-net over F2, using
(171−24, 171, 3481)-Net over F2 — Digital
Digital (147, 171, 3481)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2171, 3481, F2, 4, 24) (dual of [(3481, 4), 13753, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2171, 4100, F2, 4, 24) (dual of [(4100, 4), 16229, 25]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2171, 16400, F2, 24) (dual of [16400, 16229, 25]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2169, 16398, F2, 24) (dual of [16398, 16229, 25]-code), using
- 1 times truncation [i] based on linear OA(2170, 16399, F2, 25) (dual of [16399, 16229, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2169, 16384, F2, 25) (dual of [16384, 16215, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2170, 16399, F2, 25) (dual of [16399, 16229, 26]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2169, 16398, F2, 24) (dual of [16398, 16229, 25]-code), using
- OOA 4-folding [i] based on linear OA(2171, 16400, F2, 24) (dual of [16400, 16229, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(2171, 4100, F2, 4, 24) (dual of [(4100, 4), 16229, 25]-NRT-code), using
(171−24, 171, 103030)-Net in Base 2 — Upper bound on s
There is no (147, 171, 103031)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2993 388333 863892 798138 730276 873360 207790 063239 467082 > 2171 [i]