Best Known (131−23, 131, s)-Nets in Base 2
(131−23, 131, 320)-Net over F2 — Constructive and digital
Digital (108, 131, 320)-net over F2, using
- 21 times duplication [i] based on digital (107, 130, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 26, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 26, 64)-net over F32, using
(131−23, 131, 698)-Net over F2 — Digital
Digital (108, 131, 698)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2131, 698, F2, 2, 23) (dual of [(698, 2), 1265, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2131, 1039, F2, 2, 23) (dual of [(1039, 2), 1947, 24]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2130, 1039, F2, 2, 23) (dual of [(1039, 2), 1948, 24]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2128, 1038, F2, 2, 23) (dual of [(1038, 2), 1948, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2128, 2076, F2, 23) (dual of [2076, 1948, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2122, 2048, F2, 23) (dual of [2048, 1926, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2100, 2048, F2, 19) (dual of [2048, 1948, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(2128, 2076, F2, 23) (dual of [2076, 1948, 24]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2128, 1038, F2, 2, 23) (dual of [(1038, 2), 1948, 24]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2130, 1039, F2, 2, 23) (dual of [(1039, 2), 1948, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2131, 1039, F2, 2, 23) (dual of [(1039, 2), 1947, 24]-NRT-code), using
(131−23, 131, 17711)-Net in Base 2 — Upper bound on s
There is no (108, 131, 17712)-net in base 2, because
- 1 times m-reduction [i] would yield (108, 130, 17712)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1361 757288 101193 117858 383620 028347 255279 > 2130 [i]