Best Known (138−17, 138, s)-Nets in Base 2
(138−17, 138, 16386)-Net over F2 — Constructive and digital
Digital (121, 138, 16386)-net over F2, using
- net defined by OOA [i] based on linear OOA(2138, 16386, F2, 17, 17) (dual of [(16386, 17), 278424, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2138, 131089, F2, 17) (dual of [131089, 130951, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2138, 131090, F2, 17) (dual of [131090, 130952, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2137, 131072, F2, 17) (dual of [131072, 130935, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(2138, 131090, F2, 17) (dual of [131090, 130952, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2138, 131089, F2, 17) (dual of [131089, 130951, 18]-code), using
(138−17, 138, 21848)-Net over F2 — Digital
Digital (121, 138, 21848)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2138, 21848, F2, 6, 17) (dual of [(21848, 6), 130950, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2138, 131088, F2, 17) (dual of [131088, 130950, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2138, 131090, F2, 17) (dual of [131090, 130952, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2137, 131072, F2, 17) (dual of [131072, 130935, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(2138, 131090, F2, 17) (dual of [131090, 130952, 18]-code), using
- OOA 6-folding [i] based on linear OA(2138, 131088, F2, 17) (dual of [131088, 130950, 18]-code), using
(138−17, 138, 538046)-Net in Base 2 — Upper bound on s
There is no (121, 138, 538047)-net in base 2, because
- 1 times m-reduction [i] would yield (121, 137, 538047)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 174226 995911 937437 926303 289265 137040 380305 > 2137 [i]