Best Known (25, 44, s)-Nets in Base 27
(25, 44, 176)-Net over F27 — Constructive and digital
Digital (25, 44, 176)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 8, 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 (4, 13, 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 (2, 8, 48)-net over F27, using
(25, 44, 250)-Net in Base 27 — Constructive
(25, 44, 250)-net in base 27, using
- base change [i] based on digital (14, 33, 250)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (4, 23, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- digital (1, 10, 100)-net over F81, using
- (u, u+v)-construction [i] based on
(25, 44, 937)-Net over F27 — Digital
Digital (25, 44, 937)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2744, 937, F27, 19) (dual of [937, 893, 20]-code), using
- 198 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 6 times 0, 1, 20 times 0, 1, 54 times 0, 1, 112 times 0) [i] based on linear OA(2737, 732, F27, 19) (dual of [732, 695, 20]-code), using
- construction XX applied to C1 = C([727,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([727,17]) [i] based on
- linear OA(2735, 728, F27, 18) (dual of [728, 693, 19]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2735, 728, F27, 18) (dual of [728, 693, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2737, 728, F27, 19) (dual of [728, 691, 20]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2733, 728, F27, 17) (dual of [728, 695, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 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([727,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([727,17]) [i] based on
- 198 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 6 times 0, 1, 20 times 0, 1, 54 times 0, 1, 112 times 0) [i] based on linear OA(2737, 732, F27, 19) (dual of [732, 695, 20]-code), using
(25, 44, 1100308)-Net in Base 27 — Upper bound on s
There is no (25, 44, 1100309)-net in base 27, because
- 1 times m-reduction [i] would yield (25, 43, 1100309)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 35 370770 908440 990569 025553 457357 174586 735719 254154 977820 551491 > 2743 [i]