Best Known (8, 8+11, s)-Nets in Base 27
(8, 8+11, 86)-Net over F27 — Constructive and digital
Digital (8, 19, 86)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- digital (2, 13, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- digital (1, 6, 38)-net over F27, using
(8, 8+11, 108)-Net over F27 — Digital
Digital (8, 19, 108)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2719, 108, F27, 11) (dual of [108, 89, 12]-code), using
- construction XX applied to C1 = C([103,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([103,9]) [i] based on
- linear OA(2717, 104, F27, 10) (dual of [104, 87, 11]-code), using the BCH-code C(I) with length 104 | 272−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2717, 104, F27, 10) (dual of [104, 87, 11]-code), using the expurgated narrow-sense BCH-code C(I) with length 104 | 272−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2719, 104, F27, 11) (dual of [104, 85, 12]-code), using the BCH-code C(I) with length 104 | 272−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2715, 104, F27, 9) (dual of [104, 89, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 104 | 272−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 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
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([103,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([103,9]) [i] based on
(8, 8+11, 116)-Net in Base 27 — Constructive
(8, 19, 116)-net in base 27, using
- 5 times m-reduction [i] based on (8, 24, 116)-net in base 27, using
- base change [i] based on digital (2, 18, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 18, 116)-net over F81, using
(8, 8+11, 136)-Net in Base 27
(8, 19, 136)-net in base 27, using
- 1 times m-reduction [i] based on (8, 20, 136)-net in base 27, using
- base change [i] based on digital (3, 15, 136)-net over F81, using
- net from sequence [i] based on digital (3, 135)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 3 and N(F) ≥ 136, using
- net from sequence [i] based on digital (3, 135)-sequence over F81, using
- base change [i] based on digital (3, 15, 136)-net over F81, using
(8, 8+11, 14246)-Net in Base 27 — Upper bound on s
There is no (8, 19, 14247)-net in base 27, because
- 1 times m-reduction [i] would yield (8, 18, 14247)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 58 161649 872817 447971 361231 > 2718 [i]