Best Known (23, 41, s)-Nets in Base 27
(23, 41, 164)-Net over F27 — Constructive and digital
Digital (23, 41, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 86)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- digital (2, 11, 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 (1, 5, 38)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 25, 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 (7, 16, 86)-net over F27, using
(23, 41, 216)-Net in Base 27 — Constructive
(23, 41, 216)-net in base 27, using
- 1 times m-reduction [i] based on (23, 42, 216)-net in base 27, using
- (u, u+v)-construction [i] based on
- (5, 14, 100)-net in base 27, using
- 2 times m-reduction [i] based on (5, 16, 100)-net in base 27, using
- base change [i] based on digital (1, 12, 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
- base change [i] based on digital (1, 12, 100)-net over F81, using
- 2 times m-reduction [i] based on (5, 16, 100)-net in base 27, using
- (9, 28, 116)-net in base 27, using
- base change [i] based on digital (2, 21, 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, 21, 116)-net over F81, using
- (5, 14, 100)-net in base 27, using
- (u, u+v)-construction [i] based on
(23, 41, 838)-Net over F27 — Digital
Digital (23, 41, 838)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2741, 838, F27, 18) (dual of [838, 797, 19]-code), using
- 100 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 24 times 0, 1, 63 times 0) [i] based on linear OA(2735, 732, F27, 18) (dual of [732, 697, 19]-code), using
- construction XX applied to C1 = C([727,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([727,16]) [i] based on
- linear OA(2733, 728, F27, 17) (dual of [728, 695, 18]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [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(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(2731, 728, F27, 16) (dual of [728, 697, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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([727,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([727,16]) [i] based on
- 100 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 24 times 0, 1, 63 times 0) [i] based on linear OA(2735, 732, F27, 18) (dual of [732, 697, 19]-code), using
(23, 41, 528970)-Net in Base 27 — Upper bound on s
There is no (23, 41, 528971)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 48519 462566 084154 101945 797825 440674 971649 168231 067795 579791 > 2741 [i]