Best Known (45−11, 45, s)-Nets in Base 2
(45−11, 45, 84)-Net over F2 — Constructive and digital
Digital (34, 45, 84)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 45)-net over F2, using
- digital (21, 32, 42)-net over F2, using
- (u, u+v)-construction [i] based on
(45−11, 45, 138)-Net over F2 — Digital
Digital (34, 45, 138)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(245, 138, F2, 2, 11) (dual of [(138, 2), 231, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(245, 276, F2, 11) (dual of [276, 231, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(244, 275, F2, 11) (dual of [275, 231, 12]-code), using
- adding a parity check bit [i] based on linear OA(243, 274, F2, 10) (dual of [274, 231, 11]-code), using
- construction XX applied to C1 = C([253,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([253,8]) [i] based on
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(232, 255, F2, 8) (dual of [255, 223, 9]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(241, 255, F2, 11) (dual of [255, 214, 12]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(224, 255, F2, 6) (dual of [255, 231, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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 (see above)
- construction XX applied to C1 = C([253,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([253,8]) [i] based on
- adding a parity check bit [i] based on linear OA(243, 274, F2, 10) (dual of [274, 231, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(244, 275, F2, 11) (dual of [275, 231, 12]-code), using
- OOA 2-folding [i] based on linear OA(245, 276, F2, 11) (dual of [276, 231, 12]-code), using
(45−11, 45, 1154)-Net in Base 2 — Upper bound on s
There is no (34, 45, 1155)-net in base 2, because
- 1 times m-reduction [i] would yield (34, 44, 1155)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 17 652885 481232 > 244 [i]