Best Known (121−19, 121, s)-Nets in Base 2
(121−19, 121, 911)-Net over F2 — Constructive and digital
Digital (102, 121, 911)-net over F2, using
- 22 times duplication [i] based on digital (100, 119, 911)-net over F2, using
- net defined by OOA [i] based on linear OOA(2119, 911, F2, 19, 19) (dual of [(911, 19), 17190, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2119, 8200, F2, 19) (dual of [8200, 8081, 20]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2119, 8206, F2, 19) (dual of [8206, 8087, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2119, 8200, F2, 19) (dual of [8200, 8081, 20]-code), using
- net defined by OOA [i] based on linear OOA(2119, 911, F2, 19, 19) (dual of [(911, 19), 17190, 20]-NRT-code), using
(121−19, 121, 1963)-Net over F2 — Digital
Digital (102, 121, 1963)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2121, 1963, F2, 4, 19) (dual of [(1963, 4), 7731, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2121, 2052, F2, 4, 19) (dual of [(2052, 4), 8087, 20]-NRT-code), using
- OOA 4-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 4-folding [i] based on linear OA(2121, 8208, F2, 19) (dual of [8208, 8087, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(2121, 2052, F2, 4, 19) (dual of [(2052, 4), 8087, 20]-NRT-code), using
(121−19, 121, 42791)-Net in Base 2 — Upper bound on s
There is no (102, 121, 42792)-net in base 2, because
- 1 times m-reduction [i] would yield (102, 120, 42792)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 329496 869092 104896 428461 300032 589300 > 2120 [i]