Best Known (193−38, 193, s)-Nets in Base 3
(193−38, 193, 688)-Net over F3 — Constructive and digital
Digital (155, 193, 688)-net over F3, using
- t-expansion [i] based on digital (154, 193, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (154, 196, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 49, 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, 49, 172)-net over F81, using
- 3 times m-reduction [i] based on digital (154, 196, 688)-net over F3, using
(193−38, 193, 2358)-Net over F3 — Digital
Digital (155, 193, 2358)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3193, 2358, F3, 38) (dual of [2358, 2165, 39]-code), using
- 147 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 21 times 0, 1, 26 times 0, 1, 33 times 0) [i] based on linear OA(3176, 2194, F3, 38) (dual of [2194, 2018, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(37) ⊂ Ce(36) [i] based on
- 147 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 21 times 0, 1, 26 times 0, 1, 33 times 0) [i] based on linear OA(3176, 2194, F3, 38) (dual of [2194, 2018, 39]-code), using
(193−38, 193, 278422)-Net in Base 3 — Upper bound on s
There is no (155, 193, 278423)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 121 456457 607401 570167 548370 951562 562008 563313 784363 154195 167212 998340 506615 486450 380235 883579 > 3193 [i]