Best Known (136−23, 136, s)-Nets in Base 2
(136−23, 136, 373)-Net over F2 — Constructive and digital
Digital (113, 136, 373)-net over F2, using
- 22 times duplication [i] based on digital (111, 134, 373)-net over F2, using
- net defined by OOA [i] based on linear OOA(2134, 373, F2, 23, 23) (dual of [(373, 23), 8445, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2134, 4104, F2, 23) (dual of [4104, 3970, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2134, 4109, F2, 23) (dual of [4109, 3975, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2133, 4096, F2, 23) (dual of [4096, 3963, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2134, 4109, F2, 23) (dual of [4109, 3975, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2134, 4104, F2, 23) (dual of [4104, 3970, 24]-code), using
- net defined by OOA [i] based on linear OOA(2134, 373, F2, 23, 23) (dual of [(373, 23), 8445, 24]-NRT-code), using
(136−23, 136, 1027)-Net over F2 — Digital
Digital (113, 136, 1027)-net over F2, using
- 22 times duplication [i] based on digital (111, 134, 1027)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2134, 1027, F2, 4, 23) (dual of [(1027, 4), 3974, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2134, 4108, F2, 23) (dual of [4108, 3974, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2134, 4109, F2, 23) (dual of [4109, 3975, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2133, 4096, F2, 23) (dual of [4096, 3963, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2134, 4109, F2, 23) (dual of [4109, 3975, 24]-code), using
- OOA 4-folding [i] based on linear OA(2134, 4108, F2, 23) (dual of [4108, 3974, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2134, 1027, F2, 4, 23) (dual of [(1027, 4), 3974, 24]-NRT-code), using
(136−23, 136, 24276)-Net in Base 2 — Upper bound on s
There is no (113, 136, 24277)-net in base 2, because
- 1 times m-reduction [i] would yield (113, 135, 24277)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 43564 827951 575166 424301 820860 497905 969152 > 2135 [i]