Best Known (50, 50+45, s)-Nets in Base 27
(50, 50+45, 202)-Net over F27 — Constructive and digital
Digital (50, 95, 202)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 32, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (18, 63, 108)-net over F27, using
- net from sequence [i] based on digital (18, 107)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 18 and N(F) ≥ 108, using
- F3 from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F27 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 18 and N(F) ≥ 108, using
- net from sequence [i] based on digital (18, 107)-sequence over F27, using
- digital (10, 32, 94)-net over F27, using
(50, 50+45, 370)-Net in Base 27 — Constructive
(50, 95, 370)-net in base 27, using
- t-expansion [i] based on (43, 95, 370)-net in base 27, using
- 13 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
- 13 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
(50, 50+45, 843)-Net over F27 — Digital
Digital (50, 95, 843)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2795, 843, F27, 45) (dual of [843, 748, 46]-code), using
- 100 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 5 times 0, 1, 14 times 0, 1, 28 times 0, 1, 46 times 0) [i] based on linear OA(2787, 735, F27, 45) (dual of [735, 648, 46]-code), using
- construction XX applied to C1 = C([726,41]), C2 = C([0,42]), C3 = C1 + C2 = C([0,41]), and C∩ = C1 ∩ C2 = C([726,42]) [i] based on
- linear OA(2784, 728, F27, 44) (dual of [728, 644, 45]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,41}, and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(2782, 728, F27, 43) (dual of [728, 646, 44]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,42], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(2786, 728, F27, 45) (dual of [728, 642, 46]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,42}, and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(2780, 728, F27, 42) (dual of [728, 648, 43]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,41], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- Reed–Solomon code RS(26,27) [i]
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- 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
- construction XX applied to C1 = C([726,41]), C2 = C([0,42]), C3 = C1 + C2 = C([0,41]), and C∩ = C1 ∩ C2 = C([726,42]) [i] based on
- 100 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 5 times 0, 1, 14 times 0, 1, 28 times 0, 1, 46 times 0) [i] based on linear OA(2787, 735, F27, 45) (dual of [735, 648, 46]-code), using
(50, 50+45, 454669)-Net in Base 27 — Upper bound on s
There is no (50, 95, 454670)-net in base 27, because
- 1 times m-reduction [i] would yield (50, 94, 454670)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 353 349172 189258 003626 627469 526294 932812 480144 420264 400599 645517 053068 503151 746044 542749 125732 909770 231711 584632 764150 365230 424014 998757 > 2794 [i]