Best Known (43, 82, s)-Nets in Base 27
(43, 82, 192)-Net over F27 — Constructive and digital
Digital (43, 82, 192)-net over F27, using
- 3 times m-reduction [i] based on digital (43, 85, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 32, 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, 53, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 32, 96)-net over F27, using
- (u, u+v)-construction [i] based on
(43, 82, 370)-Net in Base 27 — Constructive
(43, 82, 370)-net in base 27, using
- 26 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
(43, 82, 749)-Net over F27 — Digital
Digital (43, 82, 749)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2782, 749, F27, 39) (dual of [749, 667, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(2782, 755, F27, 39) (dual of [755, 673, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(29) [i] based on
- linear OA(2774, 729, F27, 39) (dual of [729, 655, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2756, 729, F27, 30) (dual of [729, 673, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(278, 26, F27, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,27)), using
- discarding factors / shortening the dual code based on linear OA(278, 27, F27, 8) (dual of [27, 19, 9]-code or 27-arc in PG(7,27)), using
- Reed–Solomon code RS(19,27) [i]
- discarding factors / shortening the dual code based on linear OA(278, 27, F27, 8) (dual of [27, 19, 9]-code or 27-arc in PG(7,27)), using
- construction X applied to Ce(38) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(2782, 755, F27, 39) (dual of [755, 673, 40]-code), using
(43, 82, 385798)-Net in Base 27 — Upper bound on s
There is no (43, 82, 385799)-net in base 27, because
- 1 times m-reduction [i] would yield (43, 81, 385799)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 87 190258 613992 121300 311236 207750 447104 538850 326762 940240 306822 861943 168137 758503 149988 458833 155174 835095 321288 323187 > 2781 [i]