Best Known (121−18, 121, s)-Nets in Base 2
(121−18, 121, 912)-Net over F2 — Constructive and digital
Digital (103, 121, 912)-net over F2, using
- net defined by OOA [i] based on linear OOA(2121, 912, F2, 18, 18) (dual of [(912, 18), 16295, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2121, 8208, F2, 18) (dual of [8208, 8087, 19]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2118, 8205, F2, 18) (dual of [8205, 8087, 19]-code), using
- 1 times truncation [i] based on linear OA(2119, 8206, F2, 19) (dual of [8206, 8087, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [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(2105, 8192, F2, 17) (dual of [8192, 8087, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(18) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(2119, 8206, F2, 19) (dual of [8206, 8087, 20]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2118, 8205, F2, 18) (dual of [8205, 8087, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2121, 8208, F2, 18) (dual of [8208, 8087, 19]-code), using
(121−18, 121, 2063)-Net over F2 — Digital
Digital (103, 121, 2063)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2121, 2063, F2, 3, 18) (dual of [(2063, 3), 6068, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2121, 2736, F2, 3, 18) (dual of [(2736, 3), 8087, 19]-NRT-code), using
- strength reduction [i] based on linear OOA(2121, 2736, F2, 3, 19) (dual of [(2736, 3), 8087, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2121, 8208, F2, 19) (dual of [8208, 8087, 20]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2119, 8206, F2, 19) (dual of [8206, 8087, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [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(2105, 8192, F2, 17) (dual of [8192, 8087, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(18) ⊂ Ce(16) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2119, 8206, F2, 19) (dual of [8206, 8087, 20]-code), using
- OOA 3-folding [i] based on linear OA(2121, 8208, F2, 19) (dual of [8208, 8087, 20]-code), using
- strength reduction [i] based on linear OOA(2121, 2736, F2, 3, 19) (dual of [(2736, 3), 8087, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2121, 2736, F2, 3, 18) (dual of [(2736, 3), 8087, 19]-NRT-code), using
(121−18, 121, 46217)-Net in Base 2 — Upper bound on s
There is no (103, 121, 46218)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 658498 486010 143469 068778 604387 799475 > 2121 [i]