Best Known (176, 176+37, s)-Nets in Base 3
(176, 176+37, 1480)-Net over F3 — Constructive and digital
Digital (176, 213, 1480)-net over F3, using
- 31 times duplication [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
(176, 176+37, 5364)-Net over F3 — Digital
Digital (176, 213, 5364)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3213, 5364, F3, 37) (dual of [5364, 5151, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3213, 6621, F3, 37) (dual of [6621, 6408, 38]-code), using
- 4 times code embedding in larger space [i] based on linear OA(3209, 6617, F3, 37) (dual of [6617, 6408, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(316, 56, F3, 7) (dual of [56, 40, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to Ce(36) ⊂ Ce(28) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(3209, 6617, F3, 37) (dual of [6617, 6408, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3213, 6621, F3, 37) (dual of [6621, 6408, 38]-code), using
(176, 176+37, 1572256)-Net in Base 3 — Upper bound on s
There is no (176, 213, 1572257)-net in base 3, because
- 1 times m-reduction [i] would yield (176, 212, 1572257)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 141159 326210 619335 465775 227868 969242 437939 632737 053492 818818 233414 420260 496277 531240 361230 448224 355721 > 3212 [i]