Best Known (108, 108+18, s)-Nets in Base 2
(108, 108+18, 1820)-Net over F2 — Constructive and digital
Digital (108, 126, 1820)-net over F2, using
- net defined by OOA [i] based on linear OOA(2126, 1820, F2, 18, 18) (dual of [(1820, 18), 32634, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2126, 16380, F2, 18) (dual of [16380, 16254, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2126, 16383, F2, 18) (dual of [16383, 16257, 19]-code), using
- 1 times truncation [i] based on linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 1 times truncation [i] based on linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2126, 16383, F2, 18) (dual of [16383, 16257, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2126, 16380, F2, 18) (dual of [16380, 16254, 19]-code), using
(108, 108+18, 3770)-Net over F2 — Digital
Digital (108, 126, 3770)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2126, 3770, F2, 4, 18) (dual of [(3770, 4), 14954, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2126, 4095, F2, 4, 18) (dual of [(4095, 4), 16254, 19]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2126, 16380, F2, 18) (dual of [16380, 16254, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2126, 16383, F2, 18) (dual of [16383, 16257, 19]-code), using
- 1 times truncation [i] based on linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 1 times truncation [i] based on linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2126, 16383, F2, 18) (dual of [16383, 16257, 19]-code), using
- OOA 4-folding [i] based on linear OA(2126, 16380, F2, 18) (dual of [16380, 16254, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(2126, 4095, F2, 4, 18) (dual of [(4095, 4), 16254, 19]-NRT-code), using
(108, 108+18, 67934)-Net in Base 2 — Upper bound on s
There is no (108, 126, 67935)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 85 079906 975051 661985 674400 962419 241848 > 2126 [i]