Best Known (21, 42, s)-Nets in Base 27
(21, 42, 146)-Net over F27 — Constructive and digital
Digital (21, 42, 146)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 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 (7, 28, 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 (4, 14, 64)-net over F27, using
(21, 42, 172)-Net in Base 27 — Constructive
(21, 42, 172)-net in base 27, using
- 14 times m-reduction [i] based on (21, 56, 172)-net in base 27, using
- base change [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
(21, 42, 367)-Net over F27 — Digital
Digital (21, 42, 367)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2742, 367, F27, 2, 21) (dual of [(367, 2), 692, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2742, 734, F27, 21) (dual of [734, 692, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2742, 735, F27, 21) (dual of [735, 693, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(2741, 730, F27, 21) (dual of [730, 689, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(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([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2742, 735, F27, 21) (dual of [735, 693, 22]-code), using
- OOA 2-folding [i] based on linear OA(2742, 734, F27, 21) (dual of [734, 692, 22]-code), using
(21, 42, 128699)-Net in Base 27 — Upper bound on s
There is no (21, 42, 128700)-net in base 27, because
- 1 times m-reduction [i] would yield (21, 41, 128700)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 48519 994297 085444 771012 942920 557968 253235 528803 838562 859401 > 2741 [i]