Best Known (125−24, 125, s)-Nets in Base 3
(125−24, 125, 688)-Net over F3 — Constructive and digital
Digital (101, 125, 688)-net over F3, using
- 31 times duplication [i] based on digital (100, 124, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
(125−24, 125, 2193)-Net over F3 — Digital
Digital (101, 125, 2193)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3125, 2193, F3, 24) (dual of [2193, 2068, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, 2228, F3, 24) (dual of [2228, 2103, 25]-code), using
- 4 times code embedding in larger space [i] based on linear OA(3121, 2224, F3, 24) (dual of [2224, 2103, 25]-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(38, 36, F3, 4) (dual of [36, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-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(3121, 2224, F3, 24) (dual of [2224, 2103, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, 2228, F3, 24) (dual of [2228, 2103, 25]-code), using
(125−24, 125, 246788)-Net in Base 3 — Upper bound on s
There is no (101, 125, 246789)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 436692 166676 093251 224019 208946 125163 321365 097970 555327 543089 > 3125 [i]