Best Known (37, 66, s)-Nets in Base 27
(37, 66, 204)-Net over F27 — Constructive and digital
Digital (37, 66, 204)-net over F27, using
- generalized (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 (4, 18, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (6, 35, 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
(37, 66, 370)-Net in Base 27 — Constructive
(37, 66, 370)-net in base 27, using
- 18 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
- base change [i] based on digital (16, 63, 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, 63, 370)-net over F81, using
(37, 66, 1044)-Net over F27 — Digital
Digital (37, 66, 1044)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2766, 1044, F27, 29) (dual of [1044, 978, 30]-code), using
- 302 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 15 times 0, 1, 30 times 0, 1, 52 times 0, 1, 80 times 0, 1, 106 times 0) [i] based on linear OA(2754, 730, F27, 29) (dual of [730, 676, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(2754, 729, F27, 29) (dual of [729, 675, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2753, 729, F27, 28) (dual of [729, 676, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(270, 1, F27, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- 302 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 15 times 0, 1, 30 times 0, 1, 52 times 0, 1, 80 times 0, 1, 106 times 0) [i] based on linear OA(2754, 730, F27, 29) (dual of [730, 676, 30]-code), using
(37, 66, 1028250)-Net in Base 27 — Upper bound on s
There is no (37, 66, 1028251)-net in base 27, because
- 1 times m-reduction [i] would yield (37, 65, 1028251)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 1093 070329 413538 691829 748133 667159 268213 813576 680199 799920 049186 772199 691785 697322 114564 656109 > 2765 [i]