Best Known (44, 44+36, s)-Nets in Base 27
(44, 44+36, 204)-Net over F27 — Constructive and digital
Digital (44, 80, 204)-net over F27, using
- 1 times m-reduction [i] based on digital (44, 81, 204)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 16, 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 (4, 22, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (6, 43, 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, 16, 64)-net over F27, using
- generalized (u, u+v)-construction [i] based on
(44, 44+36, 370)-Net in Base 27 — Constructive
(44, 80, 370)-net in base 27, using
- t-expansion [i] based on (43, 80, 370)-net in base 27, using
- 28 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 81, 370)-net over F81, using
- 28 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(44, 44+36, 1018)-Net over F27 — Digital
Digital (44, 80, 1018)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2780, 1018, F27, 36) (dual of [1018, 938, 37]-code), using
- 937 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 16 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 30 times 0, 1, 33 times 0, 1, 36 times 0, 1, 40 times 0, 1, 45 times 0, 1, 49 times 0, 1, 54 times 0, 1, 60 times 0, 1, 65 times 0, 1, 73 times 0, 1, 80 times 0, 1, 88 times 0) [i] based on linear OA(2736, 37, F27, 36) (dual of [37, 1, 37]-code or 37-arc in PG(35,27)), using
- dual of repetition code with length 37 [i]
- 937 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 16 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0, 1, 30 times 0, 1, 33 times 0, 1, 36 times 0, 1, 40 times 0, 1, 45 times 0, 1, 49 times 0, 1, 54 times 0, 1, 60 times 0, 1, 65 times 0, 1, 73 times 0, 1, 80 times 0, 1, 88 times 0) [i] based on linear OA(2736, 37, F27, 36) (dual of [37, 1, 37]-code or 37-arc in PG(35,27)), using
(44, 44+36, 667986)-Net in Base 27 — Upper bound on s
There is no (44, 80, 667987)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 3 229319 444706 670098 554272 728827 955142 944535 528769 314214 953397 490144 171286 343705 728118 104601 374758 271073 100665 985525 > 2780 [i]