Best Known (67, 80, s)-Nets in Base 3
(67, 80, 3287)-Net over F3 — Constructive and digital
Digital (67, 80, 3287)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- a shift-net [i]
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (60, 73, 3280)-net over F3, using
- net defined by OOA [i] based on linear OOA(373, 3280, F3, 13, 13) (dual of [(3280, 13), 42567, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(373, 19681, F3, 13) (dual of [19681, 19608, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(373, 19681, F3, 13) (dual of [19681, 19608, 14]-code), using
- net defined by OOA [i] based on linear OOA(373, 3280, F3, 13, 13) (dual of [(3280, 13), 42567, 14]-NRT-code), using
- digital (1, 7, 7)-net over F3, using
(67, 80, 9855)-Net over F3 — Digital
Digital (67, 80, 9855)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(380, 9855, F3, 2, 13) (dual of [(9855, 2), 19630, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(380, 19710, F3, 13) (dual of [19710, 19630, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(346, 19683, F3, 8) (dual of [19683, 19637, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(380, 19710, F3, 13) (dual of [19710, 19630, 14]-code), using
(67, 80, 2866076)-Net in Base 3 — Upper bound on s
There is no (67, 80, 2866077)-net in base 3, because
- 1 times m-reduction [i] would yield (67, 79, 2866077)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 49 269664 143285 225298 544349 444834 871345 > 379 [i]