Best Known (35−19, 35, s)-Nets in Base 27
(35−19, 35, 116)-Net over F27 — Constructive and digital
Digital (16, 35, 116)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- digital (4, 23, 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 (3, 12, 52)-net over F27, using
(35−19, 35, 172)-Net in Base 27 — Constructive
(16, 35, 172)-net in base 27, using
- 1 times m-reduction [i] based on (16, 36, 172)-net in base 27, using
- base change [i] based on digital (7, 27, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 27, 172)-net over F81, using
(35−19, 35, 186)-Net over F27 — Digital
Digital (16, 35, 186)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2735, 186, F27, 19) (dual of [186, 151, 20]-code), using
- construction XX applied to C1 = C([181,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([181,17]) [i] based on
- linear OA(2733, 182, F27, 18) (dual of [182, 149, 19]-code), using the BCH-code C(I) with length 182 | 272−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2733, 182, F27, 18) (dual of [182, 149, 19]-code), using the expurgated narrow-sense BCH-code C(I) with length 182 | 272−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2735, 182, F27, 19) (dual of [182, 147, 20]-code), using the BCH-code C(I) with length 182 | 272−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2731, 182, F27, 17) (dual of [182, 151, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 182 | 272−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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([181,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([181,17]) [i] based on
(35−19, 35, 190)-Net in Base 27
(16, 35, 190)-net in base 27, using
- 5 times m-reduction [i] based on (16, 40, 190)-net in base 27, using
- base change [i] based on digital (6, 30, 190)-net over F81, using
- net from sequence [i] based on digital (6, 189)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 6 and N(F) ≥ 190, using
- net from sequence [i] based on digital (6, 189)-sequence over F81, using
- base change [i] based on digital (6, 30, 190)-net over F81, using
(35−19, 35, 40747)-Net in Base 27 — Upper bound on s
There is no (16, 35, 40748)-net in base 27, because
- 1 times m-reduction [i] would yield (16, 34, 40748)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 4 638426 046587 754196 524903 386684 481711 432795 518969 > 2734 [i]