Best Known (137, 154, s)-Nets in Base 2
(137, 154, 65538)-Net over F2 — Constructive and digital
Digital (137, 154, 65538)-net over F2, using
- net defined by OOA [i] based on linear OOA(2154, 65538, F2, 17, 17) (dual of [(65538, 17), 1113992, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2154, 524305, F2, 17) (dual of [524305, 524151, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2154, 524308, F2, 17) (dual of [524308, 524154, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2153, 524288, F2, 17) (dual of [524288, 524135, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2134, 524288, F2, 15) (dual of [524288, 524154, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2154, 524308, F2, 17) (dual of [524308, 524154, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2154, 524305, F2, 17) (dual of [524305, 524151, 18]-code), using
(137, 154, 87384)-Net over F2 — Digital
Digital (137, 154, 87384)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2154, 87384, F2, 6, 17) (dual of [(87384, 6), 524150, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2154, 524304, F2, 17) (dual of [524304, 524150, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2154, 524308, F2, 17) (dual of [524308, 524154, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2153, 524288, F2, 17) (dual of [524288, 524135, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2134, 524288, F2, 15) (dual of [524288, 524154, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2154, 524308, F2, 17) (dual of [524308, 524154, 18]-code), using
- OOA 6-folding [i] based on linear OA(2154, 524304, F2, 17) (dual of [524304, 524150, 18]-code), using
(137, 154, 2152218)-Net in Base 2 — Upper bound on s
There is no (137, 154, 2152219)-net in base 2, because
- 1 times m-reduction [i] would yield (137, 153, 2152219)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 11417 991875 258361 996829 594181 325247 816134 539555 > 2153 [i]