Best Known (31, 57, s)-Nets in Base 27
(31, 57, 178)-Net over F27 — Constructive and digital
Digital (31, 57, 178)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 21, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (10, 36, 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 (8, 21, 84)-net over F27, using
(31, 57, 370)-Net in Base 27 — Constructive
(31, 57, 370)-net in base 27, using
- 3 times m-reduction [i] based on (31, 60, 370)-net in base 27, using
- base change [i] based on digital (16, 45, 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, 45, 370)-net over F81, using
(31, 57, 781)-Net over F27 — Digital
Digital (31, 57, 781)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2757, 781, F27, 26) (dual of [781, 724, 27]-code), using
- 41 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 9 times 0, 1, 27 times 0) [i] based on linear OA(2752, 735, F27, 26) (dual of [735, 683, 27]-code), using
- construction XX applied to C1 = C([726,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([726,23]) [i] based on
- linear OA(2749, 728, F27, 25) (dual of [728, 679, 26]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,22}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2747, 728, F27, 24) (dual of [728, 681, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2751, 728, F27, 26) (dual of [728, 677, 27]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,23}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2745, 728, F27, 23) (dual of [728, 683, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [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([726,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([726,23]) [i] based on
- 41 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 9 times 0, 1, 27 times 0) [i] based on linear OA(2752, 735, F27, 26) (dual of [735, 683, 27]-code), using
(31, 57, 411533)-Net in Base 27 — Upper bound on s
There is no (31, 57, 411534)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 3870 229852 017249 310714 041502 242359 102970 560633 078473 052045 970167 519432 540466 936949 > 2757 [i]