Best Known (29, 29+25, s)-Nets in Base 27
(29, 29+25, 176)-Net over F27 — Constructive and digital
Digital (29, 54, 176)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 19, 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 (10, 35, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (7, 19, 82)-net over F27, using
(29, 29+25, 224)-Net in Base 27 — Constructive
(29, 54, 224)-net in base 27, using
- 10 times m-reduction [i] based on (29, 64, 224)-net in base 27, using
- base change [i] based on digital (13, 48, 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, 48, 224)-net over F81, using
(29, 29+25, 710)-Net over F27 — Digital
Digital (29, 54, 710)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2754, 710, F27, 25) (dual of [710, 656, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2754, 747, F27, 25) (dual of [747, 693, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(2749, 730, F27, 25) (dual of [730, 681, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2737, 730, F27, 19) (dual of [730, 693, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2754, 747, F27, 25) (dual of [747, 693, 26]-code), using
(29, 29+25, 426813)-Net in Base 27 — Upper bound on s
There is no (29, 54, 426814)-net in base 27, because
- 1 times m-reduction [i] would yield (29, 53, 426814)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 7282 489704 724154 671592 716549 531599 925933 104024 462324 932305 895570 817364 550617 > 2753 [i]