Best Known (216−31, 216, s)-Nets in Base 2
(216−31, 216, 1093)-Net over F2 — Constructive and digital
Digital (185, 216, 1093)-net over F2, using
- 24 times duplication [i] based on digital (181, 212, 1093)-net over F2, using
- net defined by OOA [i] based on linear OOA(2212, 1093, F2, 31, 31) (dual of [(1093, 31), 33671, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2212, 16396, F2, 31) (dual of [16396, 16184, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2212, 16399, F2, 31) (dual of [16399, 16187, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2212, 16399, F2, 31) (dual of [16399, 16187, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2212, 16396, F2, 31) (dual of [16396, 16184, 32]-code), using
- net defined by OOA [i] based on linear OOA(2212, 1093, F2, 31, 31) (dual of [(1093, 31), 33671, 32]-NRT-code), using
(216−31, 216, 3280)-Net over F2 — Digital
Digital (185, 216, 3280)-net over F2, using
- 22 times duplication [i] based on digital (183, 214, 3280)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2214, 3280, F2, 5, 31) (dual of [(3280, 5), 16186, 32]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2213, 3280, F2, 5, 31) (dual of [(3280, 5), 16187, 32]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2213, 16400, F2, 31) (dual of [16400, 16187, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2212, 16399, F2, 31) (dual of [16399, 16187, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(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(30) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2212, 16399, F2, 31) (dual of [16399, 16187, 32]-code), using
- OOA 5-folding [i] based on linear OA(2213, 16400, F2, 31) (dual of [16400, 16187, 32]-code), using
- 21 times duplication [i] based on linear OOA(2213, 3280, F2, 5, 31) (dual of [(3280, 5), 16187, 32]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2214, 3280, F2, 5, 31) (dual of [(3280, 5), 16186, 32]-NRT-code), using
(216−31, 216, 132573)-Net in Base 2 — Upper bound on s
There is no (185, 216, 132574)-net in base 2, because
- 1 times m-reduction [i] would yield (185, 215, 132574)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 52657 065437 544999 009792 269335 812803 885614 147610 281224 386564 483596 > 2215 [i]