Best Known (55−27, 55, s)-Nets in Base 27
(55−27, 55, 166)-Net over F27 — Constructive and digital
Digital (28, 55, 166)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (8, 35, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (7, 20, 82)-net over F27, using
(55−27, 55, 224)-Net in Base 27 — Constructive
(28, 55, 224)-net in base 27, using
- 5 times m-reduction [i] based on (28, 60, 224)-net in base 27, using
- base change [i] based on digital (13, 45, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- base change [i] based on digital (13, 45, 224)-net over F81, using
(55−27, 55, 472)-Net over F27 — Digital
Digital (28, 55, 472)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2755, 472, F27, 27) (dual of [472, 417, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2755, 738, F27, 27) (dual of [738, 683, 28]-code), using
- construction XX applied to C1 = C([725,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([725,23]) [i] based on
- linear OA(2751, 728, F27, 26) (dual of [728, 677, 27]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−3,−2,…,22}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2747, 728, F27, 24) (dual of [728, 681, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2753, 728, F27, 27) (dual of [728, 675, 28]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−3,−2,…,23}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2745, 728, F27, 23) (dual of [728, 683, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [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
- 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([725,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([725,23]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2755, 738, F27, 27) (dual of [738, 683, 28]-code), using
(55−27, 55, 192345)-Net in Base 27 — Upper bound on s
There is no (28, 55, 192346)-net in base 27, because
- 1 times m-reduction [i] would yield (28, 54, 192346)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 196627 864797 137602 764593 109010 570703 153975 491140 309293 487390 701118 011565 936813 > 2754 [i]