Best Known (153, 193, s)-Nets in Base 3
(153, 193, 688)-Net over F3 — Constructive and digital
Digital (153, 193, 688)-net over F3, using
- 31 times duplication [i] based on digital (152, 192, 688)-net over F3, using
- t-expansion [i] based on digital (151, 192, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 172)-net over F81, using
- t-expansion [i] based on digital (151, 192, 688)-net over F3, using
(153, 193, 1899)-Net over F3 — Digital
Digital (153, 193, 1899)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3193, 1899, F3, 40) (dual of [1899, 1706, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3193, 2223, F3, 40) (dual of [2223, 2030, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3193, 2223, F3, 40) (dual of [2223, 2030, 41]-code), using
(153, 193, 166895)-Net in Base 3 — Upper bound on s
There is no (153, 193, 166896)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 121 453099 165160 985665 447083 200341 954024 265555 010282 505050 940904 426394 889731 355673 755421 015681 > 3193 [i]