Best Known (175−20, 175, s)-Nets in Base 2
(175−20, 175, 13109)-Net over F2 — Constructive and digital
Digital (155, 175, 13109)-net over F2, using
- 22 times duplication [i] based on digital (153, 173, 13109)-net over F2, using
- t-expansion [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
- t-expansion [i] based on digital (152, 173, 13109)-net over F2, using
(175−20, 175, 26218)-Net over F2 — Digital
Digital (155, 175, 26218)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2175, 26218, F2, 5, 20) (dual of [(26218, 5), 130915, 21]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2172, 26218, F2, 5, 20) (dual of [(26218, 5), 130918, 21]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2172, 131090, F2, 20) (dual of [131090, 130918, 21]-code), using
- strength reduction [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
- strength reduction [i] based on linear OA(2172, 131090, F2, 21) (dual of [131090, 130918, 22]-code), using
- OOA 5-folding [i] based on linear OA(2172, 131090, F2, 20) (dual of [131090, 130918, 21]-code), using
- 23 times duplication [i] based on linear OOA(2172, 26218, F2, 5, 20) (dual of [(26218, 5), 130918, 21]-NRT-code), using
(175−20, 175, 839447)-Net in Base 2 — Upper bound on s
There is no (155, 175, 839448)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 47890 566019 910391 239423 054267 394801 494350 427180 765826 > 2175 [i]