Best Known (67, 67+24, s)-Nets in Base 27
(67, 67+24, 1728)-Net over F27 — Constructive and digital
Digital (67, 91, 1728)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 21, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (46, 70, 1640)-net over F27, using
- net defined by OOA [i] based on linear OOA(2770, 1640, F27, 24, 24) (dual of [(1640, 24), 39290, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(2770, 19680, F27, 24) (dual of [19680, 19610, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(2770, 19680, F27, 24) (dual of [19680, 19610, 25]-code), using
- net defined by OOA [i] based on linear OOA(2770, 1640, F27, 24, 24) (dual of [(1640, 24), 39290, 25]-NRT-code), using
- digital (9, 21, 88)-net over F27, using
(67, 67+24, 1757)-Net in Base 27 — Constructive
(67, 91, 1757)-net in base 27, using
- (u, u+v)-construction [i] based on
- (7, 19, 116)-net in base 27, using
- 1 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- base change [i] based on digital (2, 15, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 15, 116)-net over F81, using
- 1 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- digital (48, 72, 1641)-net over F27, using
- net defined by OOA [i] based on linear OOA(2772, 1641, F27, 24, 24) (dual of [(1641, 24), 39312, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(2772, 19692, F27, 24) (dual of [19692, 19620, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2772, 19694, F27, 24) (dual of [19694, 19622, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2772, 19694, F27, 24) (dual of [19694, 19622, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(2772, 19692, F27, 24) (dual of [19692, 19620, 25]-code), using
- net defined by OOA [i] based on linear OOA(2772, 1641, F27, 24, 24) (dual of [(1641, 24), 39312, 25]-NRT-code), using
- (7, 19, 116)-net in base 27, using
(67, 67+24, 167007)-Net over F27 — Digital
Digital (67, 91, 167007)-net over F27, using
(67, 67+24, large)-Net in Base 27 — Upper bound on s
There is no (67, 91, large)-net in base 27, because
- 22 times m-reduction [i] would yield (67, 69, large)-net in base 27, but