Best Known (26, 26+21, s)-Nets in Base 27
(26, 26+21, 170)-Net over F27 — Constructive and digital
Digital (26, 47, 170)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 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, 30, 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, 17, 82)-net over F27, using
(26, 26+21, 232)-Net in Base 27 — Constructive
(26, 47, 232)-net in base 27, using
- (u, u+v)-construction [i] based on
- (6, 16, 116)-net in base 27, using
- base change [i] based on digital (2, 12, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 12, 116)-net over F81, using
- (10, 31, 116)-net in base 27, using
- 1 times m-reduction [i] based on (10, 32, 116)-net in base 27, using
- base change [i] based on digital (2, 24, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81 (see above)
- base change [i] based on digital (2, 24, 116)-net over F81, using
- 1 times m-reduction [i] based on (10, 32, 116)-net in base 27, using
- (6, 16, 116)-net in base 27, using
(26, 26+21, 802)-Net over F27 — Digital
Digital (26, 47, 802)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2747, 802, F27, 21) (dual of [802, 755, 22]-code), using
- 64 step Varšamov–Edel lengthening with (ri) = (3, 1, 4 times 0, 1, 14 times 0, 1, 42 times 0) [i] based on linear OA(2741, 732, F27, 21) (dual of [732, 691, 22]-code), using
- construction XX applied to C1 = C([727,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([727,19]) [i] based on
- linear OA(2739, 728, F27, 20) (dual of [728, 689, 21]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2739, 728, F27, 20) (dual of [728, 689, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2741, 728, F27, 21) (dual of [728, 687, 22]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2737, 728, F27, 19) (dual of [728, 691, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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([727,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([727,19]) [i] based on
- 64 step Varšamov–Edel lengthening with (ri) = (3, 1, 4 times 0, 1, 14 times 0, 1, 42 times 0) [i] based on linear OA(2741, 732, F27, 21) (dual of [732, 691, 22]-code), using
(26, 26+21, 668764)-Net in Base 27 — Upper bound on s
There is no (26, 47, 668765)-net in base 27, because
- 1 times m-reduction [i] would yield (26, 46, 668765)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 696205 625702 247483 031251 885156 677871 388913 967493 681875 038041 254513 > 2746 [i]