Best Known (199−29, 199, s)-Nets in Base 2
(199−29, 199, 1171)-Net over F2 — Constructive and digital
Digital (170, 199, 1171)-net over F2, using
- 21 times duplication [i] based on digital (169, 198, 1171)-net over F2, using
- net defined by OOA [i] based on linear OOA(2198, 1171, F2, 29, 29) (dual of [(1171, 29), 33761, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2198, 16395, F2, 29) (dual of [16395, 16197, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2198, 16399, F2, 29) (dual of [16399, 16201, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2197, 16384, F2, 29) (dual of [16384, 16187, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2198, 16399, F2, 29) (dual of [16399, 16201, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2198, 16395, F2, 29) (dual of [16395, 16197, 30]-code), using
- net defined by OOA [i] based on linear OOA(2198, 1171, F2, 29, 29) (dual of [(1171, 29), 33761, 30]-NRT-code), using
(199−29, 199, 3229)-Net over F2 — Digital
Digital (170, 199, 3229)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2199, 3229, F2, 5, 29) (dual of [(3229, 5), 15946, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 3280, F2, 5, 29) (dual of [(3280, 5), 16201, 30]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2199, 16400, F2, 29) (dual of [16400, 16201, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2198, 16399, F2, 29) (dual of [16399, 16201, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2197, 16384, F2, 29) (dual of [16384, 16187, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(28) ⊂ Ce(26) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2198, 16399, F2, 29) (dual of [16399, 16201, 30]-code), using
- OOA 5-folding [i] based on linear OA(2199, 16400, F2, 29) (dual of [16400, 16201, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 3280, F2, 5, 29) (dual of [(3280, 5), 16201, 30]-NRT-code), using
(199−29, 199, 109345)-Net in Base 2 — Upper bound on s
There is no (170, 199, 109346)-net in base 2, because
- 1 times m-reduction [i] would yield (170, 198, 109346)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 401764 856359 477354 871544 976313 568843 298092 310590 241133 138736 > 2198 [i]