Best Known (202−29, 202, s)-Nets in Base 2
(202−29, 202, 1171)-Net over F2 — Constructive and digital
Digital (173, 202, 1171)-net over F2, using
- 24 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
(202−29, 202, 3280)-Net over F2 — Digital
Digital (173, 202, 3280)-net over F2, using
- 22 times duplication [i] based on digital (171, 200, 3280)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2200, 3280, F2, 5, 29) (dual of [(3280, 5), 16200, 30]-NRT-code), using
- 21 times duplication [i] 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
- 21 times duplication [i] based on linear OOA(2199, 3280, F2, 5, 29) (dual of [(3280, 5), 16201, 30]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2200, 3280, F2, 5, 29) (dual of [(3280, 5), 16200, 30]-NRT-code), using
(202−29, 202, 126858)-Net in Base 2 — Upper bound on s
There is no (173, 202, 126859)-net in base 2, because
- 1 times m-reduction [i] would yield (173, 201, 126859)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 214193 330450 039537 823717 686733 493717 616834 709601 906951 971696 > 2201 [i]