Best Known (126−21, 126, s)-Nets in Base 2
(126−21, 126, 411)-Net over F2 — Constructive and digital
Digital (105, 126, 411)-net over F2, using
- 22 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
(126−21, 126, 1057)-Net over F2 — Digital
Digital (105, 126, 1057)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2126, 1057, F2, 3, 21) (dual of [(1057, 3), 3045, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2126, 1371, F2, 3, 21) (dual of [(1371, 3), 3987, 22]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2123, 1370, F2, 3, 21) (dual of [(1370, 3), 3987, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2123, 4110, F2, 21) (dual of [4110, 3987, 22]-code), using
- 1 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
- 1 times code embedding in larger space [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- OOA 3-folding [i] based on linear OA(2123, 4110, F2, 21) (dual of [4110, 3987, 22]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2123, 1370, F2, 3, 21) (dual of [(1370, 3), 3987, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2126, 1371, F2, 3, 21) (dual of [(1371, 3), 3987, 22]-NRT-code), using
(126−21, 126, 26218)-Net in Base 2 — Upper bound on s
There is no (105, 126, 26219)-net in base 2, because
- 1 times m-reduction [i] would yield (105, 125, 26219)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 42 540123 010262 257366 906870 056295 521969 > 2125 [i]