Best Known (196, 196+50, s)-Nets in Base 3
(196, 196+50, 688)-Net over F3 — Constructive and digital
Digital (196, 246, 688)-net over F3, using
- t-expansion [i] based on digital (193, 246, 688)-net over F3, using
- 2 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
- 2 times m-reduction [i] based on digital (193, 248, 688)-net over F3, using
(196, 196+50, 2408)-Net over F3 — Digital
Digital (196, 246, 2408)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3246, 2408, F3, 50) (dual of [2408, 2162, 51]-code), using
- 200 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, 1, 43 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
- 200 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, 1, 43 times 0) [i] based on linear OA(3232, 2194, F3, 50) (dual of [2194, 1962, 51]-code), using
(196, 196+50, 252015)-Net in Base 3 — Upper bound on s
There is no (196, 246, 252016)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2354 303685 008336 222401 927019 110538 745333 538230 474583 793215 306127 896445 196458 098397 327116 380586 800444 941360 860276 647393 > 3246 [i]