Best Known (85−41, 85, s)-Nets in Base 27
(85−41, 85, 192)-Net over F27 — Constructive and digital
Digital (44, 85, 192)-net over F27, using
- 3 times m-reduction [i] based on digital (44, 88, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 33, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- digital (11, 55, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 33, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(85−41, 85, 370)-Net in Base 27 — Constructive
(44, 85, 370)-net in base 27, using
- t-expansion [i] based on (43, 85, 370)-net in base 27, using
- 23 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
- 23 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(85−41, 85, 699)-Net over F27 — Digital
Digital (44, 85, 699)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2785, 699, F27, 41) (dual of [699, 614, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(2785, 752, F27, 41) (dual of [752, 667, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(32) [i] based on
- linear OA(2778, 729, F27, 41) (dual of [729, 651, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2762, 729, F27, 33) (dual of [729, 667, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(277, 23, F27, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,27)), using
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- Reed–Solomon code RS(20,27) [i]
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- construction X applied to Ce(40) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(2785, 752, F27, 41) (dual of [752, 667, 42]-code), using
(85−41, 85, 328130)-Net in Base 27 — Upper bound on s
There is no (44, 85, 328131)-net in base 27, because
- 1 times m-reduction [i] would yield (44, 84, 328131)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 1 716166 468922 153880 898330 928595 673834 063324 061944 095316 313558 127840 523546 879076 309042 002270 671914 770978 842906 529100 346073 > 2784 [i]