Best Known (125−21, 125, s)-Nets in Base 2
(125−21, 125, 411)-Net over F2 — Constructive and digital
Digital (104, 125, 411)-net over F2, using
- 21 times duplication [i] based on digital (103, 124, 411)-net over F2, using
- net defined by OOA [i] based on linear OOA(2124, 411, F2, 21, 21) (dual of [(411, 21), 8507, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2124, 4111, F2, 21) (dual of [4111, 3987, 22]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2121, 4096, F2, 21) (dual of [4096, 3975, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(20) ⊂ Ce(18) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2124, 4111, F2, 21) (dual of [4111, 3987, 22]-code), using
- net defined by OOA [i] based on linear OOA(2124, 411, F2, 21, 21) (dual of [(411, 21), 8507, 22]-NRT-code), using
(125−21, 125, 1028)-Net over F2 — Digital
Digital (104, 125, 1028)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2125, 1028, F2, 4, 21) (dual of [(1028, 4), 3987, 22]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2125, 4112, F2, 21) (dual of [4112, 3987, 22]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2121, 4096, F2, 21) (dual of [4096, 3975, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(20) ⊂ Ce(18) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- OOA 4-folding [i] based on linear OA(2125, 4112, F2, 21) (dual of [4112, 3987, 22]-code), using
(125−21, 125, 24461)-Net in Base 2 — Upper bound on s
There is no (104, 125, 24462)-net in base 2, because
- 1 times m-reduction [i] would yield (104, 124, 24462)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 21 268136 931505 951172 979320 791262 482438 > 2124 [i]