Best Known (160, 160+30, s)-Nets in Base 3
(160, 160+30, 1480)-Net over F3 — Constructive and digital
Digital (160, 190, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (160, 192, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 48, 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, 48, 370)-net over F81, using
(160, 160+30, 9864)-Net over F3 — Digital
Digital (160, 190, 9864)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3190, 9864, F3, 2, 30) (dual of [(9864, 2), 19538, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3190, 19728, F3, 30) (dual of [19728, 19538, 31]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3190, 19729, F3, 30) (dual of [19729, 19539, 31]-code), using
- OOA 2-folding [i] based on linear OA(3190, 19728, F3, 30) (dual of [19728, 19538, 31]-code), using
(160, 160+30, 3550346)-Net in Base 3 — Upper bound on s
There is no (160, 190, 3550347)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4 498202 010834 350956 585348 830533 827756 741522 336915 469130 544890 984538 036169 230192 900617 135891 > 3190 [i]