Best Known (21, 21+23, s)-Nets in Base 27
(21, 21+23, 140)-Net over F27 — Constructive and digital
Digital (21, 44, 140)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 15, 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, 29, 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, 15, 64)-net over F27, using
(21, 21+23, 172)-Net in Base 27 — Constructive
(21, 44, 172)-net in base 27, using
- 12 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, 21+23, 276)-Net over F27 — Digital
Digital (21, 44, 276)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2744, 276, F27, 23) (dual of [276, 232, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2744, 364, F27, 23) (dual of [364, 320, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 364 | 272−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(2744, 364, F27, 23) (dual of [364, 320, 24]-code), using
(21, 21+23, 74360)-Net in Base 27 — Upper bound on s
There is no (21, 44, 74361)-net in base 27, because
- 1 times m-reduction [i] would yield (21, 43, 74361)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 35 375038 944543 177013 165747 764466 966516 705199 208725 394965 212131 > 2743 [i]