Best Known (164, 164+30, s)-Nets in Base 3
(164, 164+30, 1480)-Net over F3 — Constructive and digital
Digital (164, 194, 1480)-net over F3, using
- t-expansion [i] based on digital (163, 194, 1480)-net over F3, using
- 2 times m-reduction [i] based on 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
- trace code for nets [i] based on digital (16, 49, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (163, 196, 1480)-net over F3, using
(164, 164+30, 10956)-Net over F3 — Digital
Digital (164, 194, 10956)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3194, 10956, F3, 30) (dual of [10956, 10762, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3194, 19733, F3, 30) (dual of [19733, 19539, 31]-code), using
- 4 times code embedding in larger space [i] based on linear OA(3190, 19729, F3, 30) (dual of [19729, 19539, 31]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3181, 19684, F3, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(39, 45, F3, 4) (dual of [45, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(3190, 19729, F3, 30) (dual of [19729, 19539, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3194, 19733, F3, 30) (dual of [19733, 19539, 31]-code), using
(164, 164+30, 4758866)-Net in Base 3 — Upper bound on s
There is no (164, 194, 4758867)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 364 354118 993220 514966 502776 077762 553423 356781 583257 774444 517333 941163 512741 416672 142714 474035 > 3194 [i]