Best Known (177, 210, s)-Nets in Base 3
(177, 210, 1480)-Net over F3 — Constructive and digital
Digital (177, 210, 1480)-net over F3, using
- t-expansion [i] based on digital (175, 210, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (175, 212, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 53, 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, 53, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (175, 212, 1480)-net over F3, using
(177, 210, 10197)-Net over F3 — Digital
Digital (177, 210, 10197)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3210, 10197, F3, 33) (dual of [10197, 9987, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 19725, F3, 33) (dual of [19725, 19515, 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, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3210, 19725, F3, 33) (dual of [19725, 19515, 34]-code), using
(177, 210, 5806360)-Net in Base 3 — Upper bound on s
There is no (177, 210, 5806361)-net in base 3, because
- 1 times m-reduction [i] would yield (177, 209, 5806361)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5228 094478 399914 179880 799726 190130 482632 934699 893366 735617 443738 327313 038866 379415 306755 045948 662593 > 3209 [i]