Best Known (26−11, 26, s)-Nets in Base 27
(26−11, 26, 224)-Net over F27 — Constructive and digital
Digital (15, 26, 224)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- digital (0, 2, 28)-net over F27 (see above)
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 5, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 11, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 1, 28)-net over F27, using
(26−11, 26, 232)-Net in Base 27 — Constructive
(15, 26, 232)-net in base 27, using
- 271 times duplication [i] based on (14, 25, 232)-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
- (7, 18, 116)-net in base 27, using
- 2 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
- 2 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- digital (2, 7, 351)-net over F27, using
- (u, u+v)-construction [i] based on
(26−11, 26, 954)-Net over F27 — Digital
Digital (15, 26, 954)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2726, 954, F27, 11) (dual of [954, 928, 12]-code), using
- 217 step Varšamov–Edel lengthening with (ri) = (2, 1, 8 times 0, 1, 45 times 0, 1, 160 times 0) [i] based on linear OA(2721, 732, F27, 11) (dual of [732, 711, 12]-code), using
- construction XX applied to C1 = C([727,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([727,9]) [i] based on
- 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 = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [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(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(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [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([727,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([727,9]) [i] based on
- 217 step Varšamov–Edel lengthening with (ri) = (2, 1, 8 times 0, 1, 45 times 0, 1, 160 times 0) [i] based on linear OA(2721, 732, F27, 11) (dual of [732, 711, 12]-code), using
(26−11, 26, 1437742)-Net in Base 27 — Upper bound on s
There is no (15, 26, 1437743)-net in base 27, because
- 1 times m-reduction [i] would yield (15, 25, 1437743)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 608268 202279 267198 536867 741616 658655 > 2725 [i]