Best Known (214−44, 214, s)-Nets in Base 3
(214−44, 214, 688)-Net over F3 — Constructive and digital
Digital (170, 214, 688)-net over F3, using
- t-expansion [i] based on digital (169, 214, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (169, 216, 688)-net over F3, using
(214−44, 214, 2131)-Net over F3 — Digital
Digital (170, 214, 2131)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3214, 2131, F3, 44) (dual of [2131, 1917, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3214, 2223, F3, 44) (dual of [2223, 2009, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(37) [i] based on
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- 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(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(43) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3214, 2223, F3, 44) (dual of [2223, 2009, 45]-code), using
(214−44, 214, 198090)-Net in Base 3 — Upper bound on s
There is no (170, 214, 198091)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 270441 261088 628191 388631 921742 421836 993626 359449 716794 069774 679227 281569 416698 614867 504519 139013 617997 > 3214 [i]