Best Known (29, 51, s)-Nets in Base 27
(29, 51, 184)-Net over F27 — Constructive and digital
Digital (29, 51, 184)-net over F27, using
- 1 times m-reduction [i] based on digital (29, 52, 184)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 10, 56)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 7, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 3, 28)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 15, 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 (4, 27, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (3, 10, 56)-net over F27, using
- generalized (u, u+v)-construction [i] based on
(29, 51, 370)-Net in Base 27 — Constructive
(29, 51, 370)-net in base 27, using
- 1 times m-reduction [i] based on (29, 52, 370)-net in base 27, using
- base change [i] based on digital (16, 39, 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, 39, 370)-net over F81, using
(29, 51, 1012)-Net over F27 — Digital
Digital (29, 51, 1012)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2751, 1012, F27, 22) (dual of [1012, 961, 23]-code), using
- 272 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 82 times 0, 1, 131 times 0) [i] based on linear OA(2743, 732, F27, 22) (dual of [732, 689, 23]-code), using
- construction XX applied to C1 = C([727,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([727,20]) [i] based on
- linear OA(2741, 728, F27, 21) (dual of [728, 687, 22]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2741, 728, F27, 21) (dual of [728, 687, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2743, 728, F27, 22) (dual of [728, 685, 23]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2739, 728, F27, 20) (dual of [728, 689, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [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([727,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([727,20]) [i] based on
- 272 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 82 times 0, 1, 131 times 0) [i] based on linear OA(2743, 732, F27, 22) (dual of [732, 689, 23]-code), using
(29, 51, 817261)-Net in Base 27 — Upper bound on s
There is no (29, 51, 817262)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 9 989808 819550 635890 812960 584538 317736 088253 212157 014724 982586 558513 375169 > 2751 [i]