Best Known (127−25, 127, s)-Nets in Base 3
(127−25, 127, 640)-Net over F3 — Constructive and digital
Digital (102, 127, 640)-net over F3, using
- t-expansion [i] based on digital (101, 127, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (101, 128, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 32, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 32, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (101, 128, 640)-net over F3, using
(127−25, 127, 1916)-Net over F3 — Digital
Digital (102, 127, 1916)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3127, 1916, F3, 25) (dual of [1916, 1789, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 2230, F3, 25) (dual of [2230, 2103, 26]-code), using
- 4 times code embedding in larger space [i] based on linear OA(3123, 2226, F3, 25) (dual of [2226, 2103, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3113, 2188, F3, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(3123, 2226, F3, 25) (dual of [2226, 2103, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3127, 2230, F3, 25) (dual of [2230, 2103, 26]-code), using
(127−25, 127, 270449)-Net in Base 3 — Upper bound on s
There is no (102, 127, 270450)-net in base 3, because
- 1 times m-reduction [i] would yield (102, 126, 270450)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 310051 618327 692876 694416 738947 754329 223179 485989 085087 173241 > 3126 [i]