Best Known (50−22, 50, s)-Nets in Base 27
(50−22, 50, 176)-Net over F27 — Constructive and digital
Digital (28, 50, 176)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 18, 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 (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 (7, 18, 82)-net over F27, using
(50−22, 50, 232)-Net in Base 27 — Constructive
(28, 50, 232)-net in base 27, using
- (u, u+v)-construction [i] based on
- (7, 18, 116)-net in base 27, using
- 2 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- base change [i] based on digital (2, 15, 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, 15, 116)-net over F81, using
- 2 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- (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
- (7, 18, 116)-net in base 27, using
(50−22, 50, 879)-Net over F27 — Digital
Digital (28, 50, 879)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2750, 879, F27, 22) (dual of [879, 829, 23]-code), using
- 140 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 82 times 0) [i] based on linear OA(2743, 732, F27, 22) (dual of [732, 689, 23]-code), using
- construction XX applied to C1 = C([727,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([727,20]) [i] based on
- 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(2741, 728, F27, 21) (dual of [728, 687, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2743, 728, F27, 22) (dual of [728, 685, 23]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [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(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([727,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([727,20]) [i] based on
- 140 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 82 times 0) [i] based on linear OA(2743, 732, F27, 22) (dual of [732, 689, 23]-code), using
(50−22, 50, 605669)-Net in Base 27 — Upper bound on s
There is no (28, 50, 605670)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 369990 501037 521944 079562 613391 466990 849759 234060 176130 351727 257483 948449 > 2750 [i]