Best Known (163, 198, s)-Nets in Base 3
(163, 198, 896)-Net over F3 — Constructive and digital
Digital (163, 198, 896)-net over F3, using
- 2 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, 198, 4609)-Net over F3 — Digital
Digital (163, 198, 4609)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3198, 4609, F3, 35) (dual of [4609, 4411, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 6606, F3, 35) (dual of [6606, 6408, 36]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3196, 6604, F3, 35) (dual of [6604, 6408, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(311, 43, F3, 5) (dual of [43, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3196, 6604, F3, 35) (dual of [6604, 6408, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 6606, F3, 35) (dual of [6606, 6408, 36]-code), using
(163, 198, 1213116)-Net in Base 3 — Upper bound on s
There is no (163, 198, 1213117)-net in base 3, because
- 1 times m-reduction [i] would yield (163, 197, 1213117)-net in base 3, but
- 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]