Best Known (123−18, 123, s)-Nets in Base 2
(123−18, 123, 913)-Net over F2 — Constructive and digital
Digital (105, 123, 913)-net over F2, using
- net defined by OOA [i] based on linear OOA(2123, 913, F2, 18, 18) (dual of [(913, 18), 16311, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2123, 8217, F2, 18) (dual of [8217, 8094, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2123, 8223, F2, 18) (dual of [8223, 8100, 19]-code), using
- 1 times truncation [i] based on linear OA(2124, 8224, F2, 19) (dual of [8224, 8100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2118, 8192, F2, 19) (dual of [8192, 8074, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2124, 8224, F2, 19) (dual of [8224, 8100, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2123, 8223, F2, 18) (dual of [8223, 8100, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2123, 8217, F2, 18) (dual of [8217, 8094, 19]-code), using
(123−18, 123, 2280)-Net over F2 — Digital
Digital (105, 123, 2280)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2123, 2280, F2, 3, 18) (dual of [(2280, 3), 6717, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2123, 2741, F2, 3, 18) (dual of [(2741, 3), 8100, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2123, 8223, F2, 18) (dual of [8223, 8100, 19]-code), using
- 1 times truncation [i] based on linear OA(2124, 8224, F2, 19) (dual of [8224, 8100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2118, 8192, F2, 19) (dual of [8192, 8074, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2124, 8224, F2, 19) (dual of [8224, 8100, 20]-code), using
- OOA 3-folding [i] based on linear OA(2123, 8223, F2, 18) (dual of [8223, 8100, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(2123, 2741, F2, 3, 18) (dual of [(2741, 3), 8100, 19]-NRT-code), using
(123−18, 123, 53916)-Net in Base 2 — Upper bound on s
There is no (105, 123, 53917)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 10 634342 621800 025273 646832 466073 803350 > 2123 [i]