Best Known (44, 44+38, s)-Nets in Base 27
(44, 44+38, 196)-Net over F27 — Constructive and digital
Digital (44, 82, 196)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 16, 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, 23, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (5, 43, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 16, 64)-net over F27, using
(44, 44+38, 370)-Net in Base 27 — Constructive
(44, 82, 370)-net in base 27, using
- t-expansion [i] based on (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
- 26 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(44, 44+38, 864)-Net over F27 — Digital
Digital (44, 82, 864)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2782, 864, F27, 38) (dual of [864, 782, 39]-code), using
- 122 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 8 times 0, 1, 18 times 0, 1, 33 times 0, 1, 52 times 0) [i] based on linear OA(2772, 732, F27, 38) (dual of [732, 660, 39]-code), using
- construction XX applied to C1 = C([727,35]), C2 = C([0,36]), C3 = C1 + C2 = C([0,35]), and C∩ = C1 ∩ C2 = C([727,36]) [i] based on
- linear OA(2770, 728, F27, 37) (dual of [728, 658, 38]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,35}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2770, 728, F27, 37) (dual of [728, 658, 38]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,36], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2772, 728, F27, 38) (dual of [728, 656, 39]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,36}, and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2768, 728, F27, 36) (dual of [728, 660, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,35], and designed minimum distance d ≥ |I|+1 = 37 [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([727,35]), C2 = C([0,36]), C3 = C1 + C2 = C([0,35]), and C∩ = C1 ∩ C2 = C([727,36]) [i] based on
- 122 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 8 times 0, 1, 18 times 0, 1, 33 times 0, 1, 52 times 0) [i] based on linear OA(2772, 732, F27, 38) (dual of [732, 660, 39]-code), using
(44, 44+38, 458878)-Net in Base 27 — Upper bound on s
There is no (44, 82, 458879)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 2354 178820 568494 773477 336264 451536 375869 876037 812548 764459 270118 295177 301172 907786 977670 496753 229465 615535 572262 300691 > 2782 [i]