Best Known (177−27, 177, s)-Nets in Base 3
(177−27, 177, 1521)-Net over F3 — Constructive and digital
Digital (150, 177, 1521)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 15, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (135, 162, 1513)-net over F3, using
- net defined by OOA [i] based on linear OOA(3162, 1513, F3, 27, 27) (dual of [(1513, 27), 40689, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3162, 19670, F3, 27) (dual of [19670, 19508, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3162, 19682, F3, 27) (dual of [19682, 19520, 28]-code), using
- 1 times truncation [i] based on 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]
- 1 times truncation [i] based on linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3162, 19682, F3, 27) (dual of [19682, 19520, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3162, 19670, F3, 27) (dual of [19670, 19508, 28]-code), using
- net defined by OOA [i] based on linear OOA(3162, 1513, F3, 27, 27) (dual of [(1513, 27), 40689, 28]-NRT-code), using
- digital (2, 15, 8)-net over F3, using
(177−27, 177, 11605)-Net over F3 — Digital
Digital (150, 177, 11605)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3177, 11605, F3, 27) (dual of [11605, 11428, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3177, 19736, F3, 27) (dual of [19736, 19559, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- 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(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3177, 19736, F3, 27) (dual of [19736, 19559, 28]-code), using
(177−27, 177, 8163253)-Net in Base 3 — Upper bound on s
There is no (150, 177, 8163254)-net in base 3, because
- 1 times m-reduction [i] would yield (150, 176, 8163254)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 940462 102973 086303 867846 282403 125709 615602 866038 779814 726672 967985 051332 185911 295925 > 3176 [i]