Best Known (161, 197, s)-Nets in Base 3
(161, 197, 896)-Net over F3 — Constructive and digital
Digital (161, 197, 896)-net over F3, using
- 31 times duplication [i] based on digital (160, 196, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 49, 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, 49, 224)-net over F81, using
(161, 197, 3778)-Net over F3 — Digital
Digital (161, 197, 3778)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3197, 3778, F3, 36) (dual of [3778, 3581, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, 6582, F3, 36) (dual of [6582, 6385, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [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(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 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([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3197, 6582, F3, 36) (dual of [6582, 6385, 37]-code), using
(161, 197, 629383)-Net in Base 3 — Upper bound on s
There is no (161, 197, 629384)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9837 818907 423596 735447 550644 221422 423677 234058 584184 686960 918863 636783 287531 432382 320682 948785 > 3197 [i]