Best Known (175−21, 175, s)-Nets in Base 2
(175−21, 175, 13109)-Net over F2 — Constructive and digital
Digital (154, 175, 13109)-net over F2, using
- 22 times duplication [i] based on digital (152, 173, 13109)-net over F2, using
- net defined by OOA [i] based on linear OOA(2173, 13109, F2, 21, 21) (dual of [(13109, 21), 275116, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2173, 131091, F2, 21) (dual of [131091, 130918, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2171, 131072, F2, 21) (dual of [131072, 130901, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2173, 131091, F2, 21) (dual of [131091, 130918, 22]-code), using
- net defined by OOA [i] based on linear OOA(2173, 13109, F2, 21, 21) (dual of [(13109, 21), 275116, 22]-NRT-code), using
(175−21, 175, 21848)-Net over F2 — Digital
Digital (154, 175, 21848)-net over F2, using
- 23 times duplication [i] based on digital (151, 172, 21848)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2172, 21848, F2, 6, 21) (dual of [(21848, 6), 130916, 22]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2172, 131088, F2, 21) (dual of [131088, 130916, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2171, 131072, F2, 21) (dual of [131072, 130901, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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
- discarding factors / shortening the dual code based on linear OA(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- OOA 6-folding [i] based on linear OA(2172, 131088, F2, 21) (dual of [131088, 130916, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2172, 21848, F2, 6, 21) (dual of [(21848, 6), 130916, 22]-NRT-code), using
(175−21, 175, 783231)-Net in Base 2 — Upper bound on s
There is no (154, 175, 783232)-net in base 2, because
- 1 times m-reduction [i] would yield (154, 174, 783232)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 23945 372379 707719 049656 793460 512915 229183 532498 743153 > 2174 [i]