Best Known (128−16, 128, s)-Nets in Base 2
(128−16, 128, 8192)-Net over F2 — Constructive and digital
Digital (112, 128, 8192)-net over F2, using
- net defined by OOA [i] based on linear OOA(2128, 8192, F2, 16, 16) (dual of [(8192, 16), 130944, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2128, 65536, F2, 16) (dual of [65536, 65408, 17]-code), using
- 1 times truncation [i] based on linear OA(2129, 65537, F2, 17) (dual of [65537, 65408, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(2129, 65537, F2, 17) (dual of [65537, 65408, 18]-code), using
- OA 8-folding and stacking [i] based on linear OA(2128, 65536, F2, 16) (dual of [65536, 65408, 17]-code), using
(128−16, 128, 13107)-Net over F2 — Digital
Digital (112, 128, 13107)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2128, 13107, F2, 5, 16) (dual of [(13107, 5), 65407, 17]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2128, 65535, F2, 16) (dual of [65535, 65407, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2128, 65536, F2, 16) (dual of [65536, 65408, 17]-code), using
- 1 times truncation [i] based on linear OA(2129, 65537, F2, 17) (dual of [65537, 65408, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(2129, 65537, F2, 17) (dual of [65537, 65408, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2128, 65536, F2, 16) (dual of [65536, 65408, 17]-code), using
- OOA 5-folding [i] based on linear OA(2128, 65535, F2, 16) (dual of [65535, 65407, 17]-code), using
(128−16, 128, 246688)-Net in Base 2 — Upper bound on s
There is no (112, 128, 246689)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 340 282575 182027 720838 449404 562540 224172 > 2128 [i]