Best Known (162, 162+30, s)-Nets in Base 3
(162, 162+30, 1480)-Net over F3 — Constructive and digital
Digital (162, 192, 1480)-net over F3, using
- t-expansion [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
(162, 162+30, 10127)-Net over F3 — Digital
Digital (162, 192, 10127)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3192, 10127, F3, 30) (dual of [10127, 9935, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 19731, F3, 30) (dual of [19731, 19539, 31]-code), using
- strength reduction [i] based on linear OA(3192, 19731, F3, 31) (dual of [19731, 19539, 32]-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(311, 47, F3, 5) (dual of [47, 36, 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 C([0,15]) ⊂ C([0,12]) [i] based on
- strength reduction [i] based on linear OA(3192, 19731, F3, 31) (dual of [19731, 19539, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3192, 19731, F3, 30) (dual of [19731, 19539, 31]-code), using
(162, 162+30, 4110429)-Net in Base 3 — Upper bound on s
There is no (162, 192, 4110430)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 40 483882 306344 471727 938020 748394 270018 188331 185032 057861 753966 320642 076558 558880 825606 817257 > 3192 [i]