Best Known (163, 197, s)-Nets in Base 3
(163, 197, 896)-Net over F3 — Constructive and digital
Digital (163, 197, 896)-net over F3, using
- 3 times m-reduction [i] based on digital (163, 200, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 50, 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, 50, 224)-net over F81, using
(163, 197, 5318)-Net over F3 — Digital
Digital (163, 197, 5318)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3197, 5318, F3, 34) (dual of [5318, 5121, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, 6598, F3, 34) (dual of [6598, 6401, 35]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(34, 36, F3, 2) (dual of [36, 32, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3197, 6598, F3, 34) (dual of [6598, 6401, 35]-code), using
(163, 197, 1213116)-Net in Base 3 — Upper bound on s
There is no (163, 197, 1213117)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9837 563242 629154 275727 970319 268593 063334 499850 712781 980876 145043 600832 188679 233485 601410 223675 > 3197 [i]