Best Known (179, 212, s)-Nets in Base 3
(179, 212, 1484)-Net over F3 — Constructive and digital
Digital (179, 212, 1484)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 16, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- 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
- digital (0, 16, 4)-net over F3, using
(179, 212, 10948)-Net over F3 — Digital
Digital (179, 212, 10948)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3212, 10948, F3, 33) (dual of [10948, 10736, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3212, 19733, F3, 33) (dual of [19733, 19521, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3210, 19731, F3, 33) (dual of [19731, 19521, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(3199, 19684, F3, 33) (dual of [19684, 19485, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3163, 19684, F3, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,16]) ⊂ C([0,13]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3210, 19731, F3, 33) (dual of [19731, 19521, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3212, 19733, F3, 33) (dual of [19733, 19521, 34]-code), using
(179, 212, 6661074)-Net in Base 3 — Upper bound on s
There is no (179, 212, 6661075)-net in base 3, because
- 1 times m-reduction [i] would yield (179, 211, 6661075)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 47052 822897 686758 088977 576887 991625 639422 975487 086739 731849 409354 548643 048394 999468 093808 947197 277921 > 3211 [i]