Best Known (243−50, 243, s)-Nets in Base 3
(243−50, 243, 688)-Net over F3 — Constructive and digital
Digital (193, 243, 688)-net over F3, using
- 5 times m-reduction [i] based on digital (193, 248, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 62, 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, 62, 172)-net over F81, using
(243−50, 243, 2290)-Net over F3 — Digital
Digital (193, 243, 2290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3243, 2290, F3, 50) (dual of [2290, 2047, 51]-code), using
- 85 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 18 times 0, 1, 24 times 0) [i] based on linear OA(3232, 2194, F3, 50) (dual of [2194, 1962, 51]-code), using
- construction X applied to Ce(49) ⊂ Ce(48) [i] based on
- linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(49) ⊂ Ce(48) [i] based on
- 85 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 18 times 0, 1, 24 times 0) [i] based on linear OA(3232, 2194, F3, 50) (dual of [2194, 1962, 51]-code), using
(243−50, 243, 220885)-Net in Base 3 — Upper bound on s
There is no (193, 243, 220886)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 87 199324 872199 176615 943428 840699 335933 121028 746245 894746 561079 337954 776499 743845 901368 686047 165640 515961 030456 962813 > 3243 [i]