Best Known (179−35, 179, s)-Nets in Base 2
(179−35, 179, 260)-Net over F2 — Constructive and digital
Digital (144, 179, 260)-net over F2, using
- 5 times m-reduction [i] based on digital (144, 184, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 46, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 46, 65)-net over F16, using
(179−35, 179, 531)-Net over F2 — Digital
Digital (144, 179, 531)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2179, 531, F2, 2, 35) (dual of [(531, 2), 883, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2179, 1062, F2, 35) (dual of [1062, 883, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(2166, 1024, F2, 35) (dual of [1024, 858, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2141, 1024, F2, 29) (dual of [1024, 883, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(213, 38, F2, 5) (dual of [38, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(2179, 1062, F2, 35) (dual of [1062, 883, 36]-code), using
(179−35, 179, 10158)-Net in Base 2 — Upper bound on s
There is no (144, 179, 10159)-net in base 2, because
- 1 times m-reduction [i] would yield (144, 178, 10159)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 383286 865676 564648 197812 257104 177581 932747 111072 257280 > 2178 [i]