Best Known (180, 180+23, s)-Nets in Base 2
(180, 180+23, 23833)-Net over F2 — Constructive and digital
Digital (180, 203, 23833)-net over F2, using
- 22 times duplication [i] based on digital (178, 201, 23833)-net over F2, using
- net defined by OOA [i] based on linear OOA(2201, 23833, F2, 23, 23) (dual of [(23833, 23), 547958, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2201, 262164, F2, 23) (dual of [262164, 261963, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2200, 262163, F2, 23) (dual of [262163, 261963, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2199, 262144, F2, 23) (dual of [262144, 261945, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2181, 262144, F2, 21) (dual of [262144, 261963, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(22) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2200, 262163, F2, 23) (dual of [262163, 261963, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2201, 262164, F2, 23) (dual of [262164, 261963, 24]-code), using
- net defined by OOA [i] based on linear OOA(2201, 23833, F2, 23, 23) (dual of [(23833, 23), 547958, 24]-NRT-code), using
(180, 180+23, 37452)-Net over F2 — Digital
Digital (180, 203, 37452)-net over F2, using
- 22 times duplication [i] based on digital (178, 201, 37452)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2201, 37452, F2, 7, 23) (dual of [(37452, 7), 261963, 24]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2201, 262164, F2, 23) (dual of [262164, 261963, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2200, 262163, F2, 23) (dual of [262163, 261963, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2199, 262144, F2, 23) (dual of [262144, 261945, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2181, 262144, F2, 21) (dual of [262144, 261963, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(22) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2200, 262163, F2, 23) (dual of [262163, 261963, 24]-code), using
- OOA 7-folding [i] based on linear OA(2201, 262164, F2, 23) (dual of [262164, 261963, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2201, 37452, F2, 7, 23) (dual of [(37452, 7), 261963, 24]-NRT-code), using
(180, 180+23, 1655828)-Net in Base 2 — Upper bound on s
There is no (180, 203, 1655829)-net in base 2, because
- 1 times m-reduction [i] would yield (180, 202, 1655829)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6 427776 578097 620186 789227 761631 395814 818099 997250 747918 940160 > 2202 [i]