Best Known (154, 154+31, s)-Nets in Base 2
(154, 154+31, 380)-Net over F2 — Constructive and digital
Digital (154, 185, 380)-net over F2, using
- t-expansion [i] based on digital (153, 185, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 37, 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, 37, 76)-net over F32, using
(154, 154+31, 1129)-Net over F2 — Digital
Digital (154, 185, 1129)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2185, 1129, F2, 3, 31) (dual of [(1129, 3), 3202, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2185, 1370, F2, 3, 31) (dual of [(1370, 3), 3925, 32]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2183, 1370, F2, 3, 31) (dual of [(1370, 3), 3927, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2183, 4110, F2, 31) (dual of [4110, 3927, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2182, 4109, F2, 31) (dual of [4109, 3927, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2181, 4096, F2, 31) (dual of [4096, 3915, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(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(30) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2182, 4109, F2, 31) (dual of [4109, 3927, 32]-code), using
- OOA 3-folding [i] based on linear OA(2183, 4110, F2, 31) (dual of [4110, 3927, 32]-code), using
- 22 times duplication [i] based on linear OOA(2183, 1370, F2, 3, 31) (dual of [(1370, 3), 3927, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2185, 1370, F2, 3, 31) (dual of [(1370, 3), 3925, 32]-NRT-code), using
(154, 154+31, 31630)-Net in Base 2 — Upper bound on s
There is no (154, 185, 31631)-net in base 2, because
- 1 times m-reduction [i] would yield (154, 184, 31631)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 24 531388 460165 944647 267454 539680 285362 026572 417453 408190 > 2184 [i]