Best Known (175, 175+38, s)-Nets in Base 3
(175, 175+38, 896)-Net over F3 — Constructive and digital
Digital (175, 213, 896)-net over F3, using
- 3 times m-reduction [i] based on digital (175, 216, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 54, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 54, 224)-net over F81, using
(175, 175+38, 4573)-Net over F3 — Digital
Digital (175, 213, 4573)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3213, 4573, F3, 38) (dual of [4573, 4360, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3213, 6605, F3, 38) (dual of [6605, 6392, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(31) [i] based on
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(312, 44, F3, 5) (dual of [44, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 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(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- 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(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to Ce(37) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3213, 6605, F3, 38) (dual of [6605, 6392, 39]-code), using
(175, 175+38, 885027)-Net in Base 3 — Upper bound on s
There is no (175, 213, 885028)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 423478 128539 884887 639594 571161 713506 967200 977589 874994 207427 018331 172492 973107 164078 385085 182966 001713 > 3213 [i]