Best Known (23−11, 23, s)-Nets in Base 27
(23−11, 23, 147)-Net over F27 — Constructive and digital
Digital (12, 23, 147)-net over F27, using
- net defined by OOA [i] based on linear OOA(2723, 147, F27, 11, 11) (dual of [(147, 11), 1594, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2723, 736, F27, 11) (dual of [736, 713, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2723, 737, F27, 11) (dual of [737, 714, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(2721, 729, F27, 11) (dual of [729, 708, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2715, 729, F27, 8) (dual of [729, 714, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(2723, 737, F27, 11) (dual of [737, 714, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2723, 736, F27, 11) (dual of [736, 713, 12]-code), using
(23−11, 23, 200)-Net in Base 27 — Constructive
(12, 23, 200)-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
- (5, 16, 100)-net in base 27, using
- base change [i] based on digital (1, 12, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- base change [i] based on digital (1, 12, 100)-net over F81, using
- digital (2, 7, 351)-net over F27, using
(23−11, 23, 499)-Net over F27 — Digital
Digital (12, 23, 499)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2723, 499, F27, 11) (dual of [499, 476, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2723, 737, F27, 11) (dual of [737, 714, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(2721, 729, F27, 11) (dual of [729, 708, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2715, 729, F27, 8) (dual of [729, 714, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(2723, 737, F27, 11) (dual of [737, 714, 12]-code), using
(23−11, 23, 199002)-Net in Base 27 — Upper bound on s
There is no (12, 23, 199003)-net in base 27, because
- 1 times m-reduction [i] would yield (12, 22, 199003)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 30 903490 359749 554509 653090 391447 > 2722 [i]