Best Known (195, 195+50, s)-Nets in Base 3
(195, 195+50, 688)-Net over F3 — Constructive and digital
Digital (195, 245, 688)-net over F3, using
- t-expansion [i] based on digital (193, 245, 688)-net over F3, using
- 3 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
- 3 times m-reduction [i] based on digital (193, 248, 688)-net over F3, using
(195, 195+50, 2363)-Net over F3 — Digital
Digital (195, 245, 2363)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3245, 2363, F3, 50) (dual of [2363, 2118, 51]-code), using
- 156 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, 1, 31 times 0, 1, 38 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
- 156 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, 1, 31 times 0, 1, 38 times 0) [i] based on linear OA(3232, 2194, F3, 50) (dual of [2194, 1962, 51]-code), using
(195, 195+50, 241179)-Net in Base 3 — Upper bound on s
There is no (195, 245, 241180)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 784 764436 051711 874895 090574 214356 300803 470832 583427 632129 909879 200093 715948 506463 288324 488488 652257 772482 510211 315897 > 3245 [i]