Best Known (199, 199+43, s)-Nets in Base 3
(199, 199+43, 1480)-Net over F3 — Constructive and digital
Digital (199, 242, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (199, 244, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 61, 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
- trace code for nets [i] based on digital (16, 61, 370)-net over F81, using
(199, 199+43, 5106)-Net over F3 — Digital
Digital (199, 242, 5106)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3242, 5106, F3, 43) (dual of [5106, 4864, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 6618, F3, 43) (dual of [6618, 6376, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(34) [i] based on
- linear OA(3225, 6561, F3, 43) (dual of [6561, 6336, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- 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(317, 57, F3, 7) (dual of [57, 40, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(42) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 6618, F3, 43) (dual of [6618, 6376, 44]-code), using
(199, 199+43, 1297149)-Net in Base 3 — Upper bound on s
There is no (199, 242, 1297150)-net in base 3, because
- 1 times m-reduction [i] would yield (199, 241, 1297150)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 9 687781 370841 901420 080377 465254 506351 644248 075746 653844 776129 696242 109364 246231 715827 726884 572100 254354 870926 340341 > 3241 [i]