Best Known (127−22, 127, s)-Nets in Base 2
(127−22, 127, 320)-Net over F2 — Constructive and digital
Digital (105, 127, 320)-net over F2, using
- 22 times duplication [i] based on digital (103, 125, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 25, 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, 25, 64)-net over F32, using
(127−22, 127, 731)-Net over F2 — Digital
Digital (105, 127, 731)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2127, 731, F2, 2, 22) (dual of [(731, 2), 1335, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2127, 1037, F2, 2, 22) (dual of [(1037, 2), 1947, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2127, 2074, F2, 22) (dual of [2074, 1947, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2127, 2075, F2, 22) (dual of [2075, 1948, 23]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2128, 2076, F2, 23) (dual of [2076, 1948, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2127, 2075, F2, 22) (dual of [2075, 1948, 23]-code), using
- OOA 2-folding [i] based on linear OA(2127, 2074, F2, 22) (dual of [2074, 1947, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(2127, 1037, F2, 2, 22) (dual of [(1037, 2), 1947, 23]-NRT-code), using
(127−22, 127, 14657)-Net in Base 2 — Upper bound on s
There is no (105, 127, 14658)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 170 167020 281909 318572 337279 565088 134592 > 2127 [i]