Best Known (193, 193+51, s)-Nets in Base 3
(193, 193+51, 688)-Net over F3 — Constructive and digital
Digital (193, 244, 688)-net over F3, using
- 4 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
(193, 193+51, 2175)-Net over F3 — Digital
Digital (193, 244, 2175)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3244, 2175, F3, 51) (dual of [2175, 1931, 52]-code), using
- discarding factors / shortening the dual code based on linear OA(3244, 2209, F3, 51) (dual of [2209, 1965, 52]-code), using
- construction XX applied to Ce(51) ⊂ Ce(48) ⊂ Ce(46) [i] based on
- linear OA(3239, 2187, F3, 52) (dual of [2187, 1948, 53]-code), using an extension Ce(51) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,51], and designed minimum distance d ≥ |I|+1 = 52 [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(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(31, 4, F3, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)
- construction XX applied to Ce(51) ⊂ Ce(48) ⊂ Ce(46) [i] based on
- discarding factors / shortening the dual code based on linear OA(3244, 2209, F3, 51) (dual of [2209, 1965, 52]-code), using
(193, 193+51, 220885)-Net in Base 3 — Upper bound on s
There is no (193, 244, 220886)-net in base 3, because
- 1 times m-reduction [i] would yield (193, 243, 220886)-net in base 3, but
- 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]