Best Known (171−29, 171, s)-Nets in Base 2
(171−29, 171, 380)-Net over F2 — Constructive and digital
Digital (142, 171, 380)-net over F2, using
- 21 times duplication [i] based on digital (141, 170, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 34, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 34, 76)-net over F32, using
(171−29, 171, 1037)-Net over F2 — Digital
Digital (142, 171, 1037)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2171, 1037, F2, 3, 29) (dual of [(1037, 3), 2940, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2171, 1370, F2, 3, 29) (dual of [(1370, 3), 3939, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2171, 4110, F2, 29) (dual of [4110, 3939, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2170, 4109, F2, 29) (dual of [4109, 3939, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2169, 4096, F2, 29) (dual of [4096, 3927, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2157, 4096, F2, 27) (dual of [4096, 3939, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(28) ⊂ Ce(26) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2170, 4109, F2, 29) (dual of [4109, 3939, 30]-code), using
- OOA 3-folding [i] based on linear OA(2171, 4110, F2, 29) (dual of [4110, 3939, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2171, 1370, F2, 3, 29) (dual of [(1370, 3), 3939, 30]-NRT-code), using
(171−29, 171, 27320)-Net in Base 2 — Upper bound on s
There is no (142, 171, 27321)-net in base 2, because
- 1 times m-reduction [i] would yield (142, 170, 27321)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1496 589935 339728 980759 684394 132344 742397 118392 547336 > 2170 [i]