Best Known (76, 76+27, s)-Nets in Base 27
(76, 76+27, 1634)-Net over F27 — Constructive and digital
Digital (76, 103, 1634)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (12, 25, 121)-net over F27, using
- net defined by OOA [i] based on linear OOA(2725, 121, F27, 13, 13) (dual of [(121, 13), 1548, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2725, 727, F27, 13) (dual of [727, 702, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2725, 727, F27, 13) (dual of [727, 702, 14]-code), using
- net defined by OOA [i] based on linear OOA(2725, 121, F27, 13, 13) (dual of [(121, 13), 1548, 14]-NRT-code), using
- digital (51, 78, 1513)-net over F27, using
- net defined by OOA [i] based on linear OOA(2778, 1513, F27, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2778, 19670, F27, 27) (dual of [19670, 19592, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19682, F27, 27) (dual of [19682, 19604, 28]-code), using
- 1 times truncation [i] based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19682, F27, 27) (dual of [19682, 19604, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2778, 19670, F27, 27) (dual of [19670, 19592, 28]-code), using
- net defined by OOA [i] based on linear OOA(2778, 1513, F27, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
- digital (12, 25, 121)-net over F27, using
(76, 76+27, 1674)-Net in Base 27 — Constructive
(76, 103, 1674)-net in base 27, using
- (u, u+v)-construction [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
- digital (52, 79, 1514)-net over F27, using
- net defined by OOA [i] based on linear OOA(2779, 1514, F27, 27, 27) (dual of [(1514, 27), 40799, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2779, 19683, F27, 27) (dual of [19683, 19604, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2779, 19684, F27, 27) (dual of [19684, 19605, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2779, 19684, F27, 27) (dual of [19684, 19605, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2779, 19683, F27, 27) (dual of [19683, 19604, 28]-code), using
- net defined by OOA [i] based on linear OOA(2779, 1514, F27, 27, 27) (dual of [(1514, 27), 40799, 28]-NRT-code), using
- (11, 24, 160)-net in base 27, using
(76, 76+27, 189999)-Net over F27 — Digital
Digital (76, 103, 189999)-net over F27, using
(76, 76+27, large)-Net in Base 27 — Upper bound on s
There is no (76, 103, large)-net in base 27, because
- 25 times m-reduction [i] would yield (76, 78, large)-net in base 27, but