Best Known (19, 38, s)-Nets in Base 27
(19, 38, 140)-Net over F27 — Constructive and digital
Digital (19, 38, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 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 (6, 25, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (4, 13, 64)-net over F27, using
(19, 38, 172)-Net in Base 27 — Constructive
(19, 38, 172)-net in base 27, using
- 10 times m-reduction [i] based on (19, 48, 172)-net in base 27, using
- base change [i] based on digital (7, 36, 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, 36, 172)-net over F81, using
(19, 38, 367)-Net over F27 — Digital
Digital (19, 38, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2738, 367, F27, 2, 19) (dual of [(367, 2), 696, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2738, 734, F27, 19) (dual of [734, 696, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2738, 735, F27, 19) (dual of [735, 697, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- 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(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(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([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2738, 735, F27, 19) (dual of [735, 697, 20]-code), using
- OOA 2-folding [i] based on linear OA(2738, 734, F27, 19) (dual of [734, 696, 20]-code), using
(19, 38, 122252)-Net in Base 27 — Upper bound on s
There is no (19, 38, 122253)-net in base 27, because
- 1 times m-reduction [i] would yield (19, 37, 122253)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 91300 209978 298806 235143 859068 136623 244412 468193 103603 > 2737 [i]