Best Known (11, 11+10, s)-Nets in Base 27
(11, 11+10, 147)-Net over F27 — Constructive and digital
Digital (11, 21, 147)-net over F27, using
- 271 times duplication [i] based on digital (10, 20, 147)-net over F27, using
- net defined by OOA [i] based on linear OOA(2720, 147, F27, 10, 10) (dual of [(147, 10), 1450, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2720, 735, F27, 10) (dual of [735, 715, 11]-code), using
- construction XX applied to C1 = C([726,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([726,7]) [i] based on
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,7}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- Reed–Solomon code RS(26,27) [i]
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- 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
- construction XX applied to C1 = C([726,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([726,7]) [i] based on
- OA 5-folding and stacking [i] based on linear OA(2720, 735, F27, 10) (dual of [735, 715, 11]-code), using
- net defined by OOA [i] based on linear OOA(2720, 147, F27, 10, 10) (dual of [(147, 10), 1450, 11]-NRT-code), using
(11, 11+10, 164)-Net in Base 27 — Constructive
(11, 21, 164)-net in base 27, using
- 1 times m-reduction [i] based on (11, 22, 164)-net in base 27, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 351)-net over F27, using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- (4, 15, 82)-net in base 27, using
- 1 times m-reduction [i] based on (4, 16, 82)-net in base 27, using
- base change [i] based on digital (0, 12, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base change [i] based on digital (0, 12, 82)-net over F81, using
- 1 times m-reduction [i] based on (4, 16, 82)-net in base 27, using
- digital (2, 7, 351)-net over F27, using
- (u, u+v)-construction [i] based on
(11, 11+10, 545)-Net over F27 — Digital
Digital (11, 21, 545)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2721, 545, F27, 10) (dual of [545, 524, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2721, 737, F27, 10) (dual of [737, 716, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(2719, 729, F27, 10) (dual of [729, 710, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2713, 729, F27, 7) (dual of [729, 716, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-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(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(2721, 737, F27, 10) (dual of [737, 716, 11]-code), using
(11, 11+10, 102939)-Net in Base 27 — Upper bound on s
There is no (11, 21, 102940)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 1 144596 881523 403261 658532 604089 > 2721 [i]