Best Known (17, 34, s)-Nets in Base 27
(17, 34, 132)-Net over F27 — Constructive and digital
Digital (17, 34, 132)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (5, 22, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 12, 64)-net over F27, using
(17, 34, 172)-Net in Base 27 — Constructive
(17, 34, 172)-net in base 27, using
- 6 times m-reduction [i] based on (17, 40, 172)-net in base 27, using
- base change [i] based on digital (7, 30, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 30, 172)-net over F81, using
(17, 34, 367)-Net over F27 — Digital
Digital (17, 34, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2734, 367, F27, 2, 17) (dual of [(367, 2), 700, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2734, 734, F27, 17) (dual of [734, 700, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2734, 735, F27, 17) (dual of [735, 701, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- 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(2729, 730, F27, 15) (dual of [730, 701, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2734, 735, F27, 17) (dual of [735, 701, 18]-code), using
- OOA 2-folding [i] based on linear OA(2734, 734, F27, 17) (dual of [734, 700, 18]-code), using
(17, 34, 116165)-Net in Base 27 — Upper bound on s
There is no (17, 34, 116166)-net in base 27, because
- 1 times m-reduction [i] would yield (17, 33, 116166)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 171795 752775 660617 239636 990474 452986 124901 839825 > 2733 [i]