Best Known (163, 163+33, s)-Nets in Base 3
(163, 163+33, 1480)-Net over F3 — Constructive and digital
Digital (163, 196, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 49, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(163, 163+33, 6197)-Net over F3 — Digital
Digital (163, 196, 6197)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3196, 6197, F3, 33) (dual of [6197, 6001, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, 6629, F3, 33) (dual of [6629, 6433, 34]-code), using
- 3 times code embedding in larger space [i] based on linear OA(3193, 6626, F3, 33) (dual of [6626, 6433, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- 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(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(316, 64, F3, 7) (dual of [64, 48, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(3193, 6626, F3, 33) (dual of [6626, 6433, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, 6629, F3, 33) (dual of [6629, 6433, 34]-code), using
(163, 163+33, 2220347)-Net in Base 3 — Upper bound on s
There is no (163, 196, 2220348)-net in base 3, because
- 1 times m-reduction [i] would yield (163, 195, 2220348)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1093 064042 764791 914487 091325 040354 518905 781781 053414 388904 221698 163021 195970 161990 594196 305537 > 3195 [i]