Best Known (32, 64, s)-Nets in Base 27
(32, 64, 170)-Net over F27 — Constructive and digital
Digital (32, 64, 170)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 23, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (9, 41, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (7, 23, 82)-net over F27, using
(32, 64, 370)-Net in Base 27 — Constructive
(32, 64, 370)-net in base 27, using
- base change [i] based on digital (16, 48, 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
(32, 64, 456)-Net over F27 — Digital
Digital (32, 64, 456)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2764, 456, F27, 32) (dual of [456, 392, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2764, 741, F27, 32) (dual of [741, 677, 33]-code), using
- construction XX applied to C1 = C([726,28]), C2 = C([3,29]), C3 = C1 + C2 = C([3,28]), and C∩ = C1 ∩ C2 = C([726,29]) [i] based on
- linear OA(2758, 728, F27, 31) (dual of [728, 670, 32]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,28}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2753, 728, F27, 27) (dual of [728, 675, 28]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {3,4,…,29}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2760, 728, F27, 32) (dual of [728, 668, 33]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,29}, and designed minimum distance d ≥ |I|+1 = 33 [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 = {3,4,…,28}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(274, 11, F27, 4) (dual of [11, 7, 5]-code or 11-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), 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([726,28]), C2 = C([3,29]), C3 = C1 + C2 = C([3,28]), and C∩ = C1 ∩ C2 = C([726,29]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2764, 741, F27, 32) (dual of [741, 677, 33]-code), using
(32, 64, 138993)-Net in Base 27 — Upper bound on s
There is no (32, 64, 138994)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 40 485221 629897 814790 051023 460928 461839 966735 033571 145117 244350 094308 957883 004064 478318 815905 > 2764 [i]