Best Known (163, 163+43, s)-Nets in Base 3
(163, 163+43, 688)-Net over F3 — Constructive and digital
Digital (163, 206, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (163, 208, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 52, 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, 52, 172)-net over F81, using
(163, 163+43, 1923)-Net over F3 — Digital
Digital (163, 206, 1923)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3206, 1923, F3, 43) (dual of [1923, 1717, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3206, 2218, F3, 43) (dual of [2218, 2012, 44]-code), using
- construction XX applied to Ce(42) ⊂ Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [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(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(38, 30, F3, 4) (dual of [30, 22, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(42) ⊂ Ce(37) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(3206, 2218, F3, 43) (dual of [2218, 2012, 44]-code), using
(163, 163+43, 197255)-Net in Base 3 — Upper bound on s
There is no (163, 206, 197256)-net in base 3, because
- 1 times m-reduction [i] would yield (163, 205, 197256)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 64 545860 530525 121531 195558 953250 657414 443783 355579 367523 036679 185810 311980 734177 746006 587937 111441 > 3205 [i]