Best Known (176, 209, s)-Nets in Base 3
(176, 209, 1480)-Net over F3 — Constructive and digital
Digital (176, 209, 1480)-net over F3, using
- t-expansion [i] based on digital (175, 209, 1480)-net over F3, using
- 3 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
- 3 times m-reduction [i] based on digital (175, 212, 1480)-net over F3, using
(176, 209, 9864)-Net over F3 — Digital
Digital (176, 209, 9864)-net over F3, using
- 31 times duplication [i] based on digital (175, 208, 9864)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3208, 9864, F3, 2, 33) (dual of [(9864, 2), 19520, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3208, 19728, F3, 33) (dual of [19728, 19520, 34]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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 Ce(33) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3208, 19728, F3, 33) (dual of [19728, 19520, 34]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3208, 9864, F3, 2, 33) (dual of [(9864, 2), 19520, 34]-NRT-code), using
(176, 209, 5421054)-Net in Base 3 — Upper bound on s
There is no (176, 209, 5421055)-net in base 3, because
- 1 times m-reduction [i] would yield (176, 208, 5421055)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1742 694339 624370 135163 362734 641938 633302 887024 176383 158475 232484 962057 298703 658219 521382 199093 690337 > 3208 [i]