Best Known (50−23, 50, s)-Nets in Base 27
(50−23, 50, 170)-Net over F27 — Constructive and digital
Digital (27, 50, 170)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 18, 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 (9, 32, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (7, 18, 82)-net over F27, using
(50−23, 50, 224)-Net in Base 27 — Constructive
(27, 50, 224)-net in base 27, using
- 6 times m-reduction [i] based on (27, 56, 224)-net in base 27, using
- base change [i] based on digital (13, 42, 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, 42, 224)-net over F81, using
(50−23, 50, 720)-Net over F27 — Digital
Digital (27, 50, 720)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2750, 720, F27, 23) (dual of [720, 670, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2750, 747, F27, 23) (dual of [747, 697, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(2745, 730, F27, 23) (dual of [730, 685, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2733, 730, F27, 17) (dual of [730, 697, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(275, 17, F27, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2750, 747, F27, 23) (dual of [747, 697, 24]-code), using
(50−23, 50, 448859)-Net in Base 27 — Upper bound on s
There is no (27, 50, 448860)-net in base 27, because
- 1 times m-reduction [i] would yield (27, 49, 448860)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 13703 414158 023515 974361 082311 193100 987683 271570 308619 302701 696455 539953 > 2749 [i]