Best Known (138−24, 138, s)-Nets in Base 3
(138−24, 138, 696)-Net over F3 — Constructive and digital
Digital (114, 138, 696)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- 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
- digital (2, 14, 8)-net over F3, using
(138−24, 138, 4216)-Net over F3 — Digital
Digital (114, 138, 4216)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3138, 4216, F3, 24) (dual of [4216, 4078, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3138, 6603, F3, 24) (dual of [6603, 6465, 25]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3137, 6602, F3, 24) (dual of [6602, 6465, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 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
- 1 times code embedding in larger space [i] based on linear OA(3137, 6602, F3, 24) (dual of [6602, 6465, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3138, 6603, F3, 24) (dual of [6603, 6465, 25]-code), using
(138−24, 138, 811371)-Net in Base 3 — Upper bound on s
There is no (114, 138, 811372)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 696204 851096 846088 306364 413022 953295 258916 052900 719322 290285 767361 > 3138 [i]