Best Known (40, 40+38, s)-Nets in Base 27
(40, 40+38, 190)-Net over F27 — Constructive and digital
Digital (40, 78, 190)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 29, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (11, 49, 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 (10, 29, 94)-net over F27, using
(40, 40+38, 370)-Net in Base 27 — Constructive
(40, 78, 370)-net in base 27, using
- 18 times m-reduction [i] based on (40, 96, 370)-net in base 27, using
- base change [i] based on digital (16, 72, 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, 72, 370)-net over F81, using
(40, 40+38, 616)-Net over F27 — Digital
Digital (40, 78, 616)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2778, 616, F27, 38) (dual of [616, 538, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 749, F27, 38) (dual of [749, 671, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- linear OA(2772, 729, F27, 38) (dual of [729, 657, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2758, 729, F27, 31) (dual of [729, 671, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(276, 20, F27, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,27)), using
- discarding factors / shortening the dual code based on linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- Reed–Solomon code RS(21,27) [i]
- discarding factors / shortening the dual code based on linear OA(276, 27, F27, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2778, 749, F27, 38) (dual of [749, 671, 39]-code), using
(40, 40+38, 229270)-Net in Base 27 — Upper bound on s
There is no (40, 78, 229271)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 4429 784248 608916 259710 253460 189960 474617 770354 876954 328487 443883 371689 132068 674433 317398 116578 780859 239112 069171 > 2778 [i]