Best Known (71, 71+25, s)-Nets in Base 27
(71, 71+25, 1762)-Net over F27 — Constructive and digital
Digital (71, 96, 1762)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 23, 122)-net over F27, using
- net defined by OOA [i] based on linear OOA(2723, 122, F27, 12, 12) (dual of [(122, 12), 1441, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2723, 732, F27, 12) (dual of [732, 709, 13]-code), using
- construction XX applied to C1 = C([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([727,10]) [i] based on
- OA 6-folding and stacking [i] based on linear OA(2723, 732, F27, 12) (dual of [732, 709, 13]-code), using
- net defined by OOA [i] based on linear OOA(2723, 122, F27, 12, 12) (dual of [(122, 12), 1441, 13]-NRT-code), using
- digital (48, 73, 1640)-net over F27, using
- net defined by OOA [i] based on linear OOA(2773, 1640, F27, 25, 25) (dual of [(1640, 25), 40927, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2773, 19681, F27, 25) (dual of [19681, 19608, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2773, 19683, F27, 25) (dual of [19683, 19610, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(2773, 19683, F27, 25) (dual of [19683, 19610, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2773, 19681, F27, 25) (dual of [19681, 19608, 26]-code), using
- net defined by OOA [i] based on linear OOA(2773, 1640, F27, 25, 25) (dual of [(1640, 25), 40927, 26]-NRT-code), using
- digital (11, 23, 122)-net over F27, using
(71, 71+25, 1800)-Net in Base 27 — Constructive
(71, 96, 1800)-net in base 27, using
- (u, u+v)-construction [i] based on
- (11, 23, 160)-net in base 27, using
- 1 times m-reduction [i] based on (11, 24, 160)-net in base 27, using
- base change [i] based on digital (5, 18, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- base change [i] based on digital (5, 18, 160)-net over F81, using
- 1 times m-reduction [i] based on (11, 24, 160)-net in base 27, using
- digital (48, 73, 1640)-net over F27, using
- net defined by OOA [i] based on linear OOA(2773, 1640, F27, 25, 25) (dual of [(1640, 25), 40927, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2773, 19681, F27, 25) (dual of [19681, 19608, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2773, 19683, F27, 25) (dual of [19683, 19610, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(2773, 19683, F27, 25) (dual of [19683, 19610, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2773, 19681, F27, 25) (dual of [19681, 19608, 26]-code), using
- net defined by OOA [i] based on linear OOA(2773, 1640, F27, 25, 25) (dual of [(1640, 25), 40927, 26]-NRT-code), using
- (11, 23, 160)-net in base 27, using
(71, 71+25, 200387)-Net over F27 — Digital
Digital (71, 96, 200387)-net over F27, using
(71, 71+25, large)-Net in Base 27 — Upper bound on s
There is no (71, 96, large)-net in base 27, because
- 23 times m-reduction [i] would yield (71, 73, large)-net in base 27, but