Best Known (194−40, 194, s)-Nets in Base 3
(194−40, 194, 688)-Net over F3 — Constructive and digital
Digital (154, 194, 688)-net over F3, using
- 2 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
(194−40, 194, 1956)-Net over F3 — Digital
Digital (154, 194, 1956)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3194, 1956, F3, 40) (dual of [1956, 1762, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3194, 2226, F3, 40) (dual of [2226, 2032, 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(311, 39, F3, 5) (dual of [39, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(37, 14, F3, 5) (dual of [14, 7, 6]-code), using
- extended quadratic residue code Qe(14,3) [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3194, 2226, F3, 40) (dual of [2226, 2032, 41]-code), using
(194−40, 194, 176321)-Net in Base 3 — Upper bound on s
There is no (154, 194, 176322)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 364 389509 529925 991588 634565 062184 508896 633408 288442 941729 186956 160048 002542 619506 540175 279561 > 3194 [i]