Best Known (55−24, 55, s)-Nets in Base 27
(55−24, 55, 182)-Net over F27 — Constructive and digital
Digital (31, 55, 182)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 21, 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 (10, 34, 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 (9, 21, 88)-net over F27, using
(55−24, 55, 370)-Net in Base 27 — Constructive
(31, 55, 370)-net in base 27, using
- 5 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
(55−24, 55, 974)-Net over F27 — Digital
Digital (31, 55, 974)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2755, 974, F27, 24) (dual of [974, 919, 25]-code), using
- 234 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 10 times 0, 1, 31 times 0, 1, 71 times 0, 1, 114 times 0) [i] based on linear OA(2747, 732, F27, 24) (dual of [732, 685, 25]-code), using
- construction XX applied to C1 = C([727,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([727,22]) [i] based on
- linear OA(2745, 728, F27, 23) (dual of [728, 683, 24]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [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(2747, 728, F27, 24) (dual of [728, 681, 25]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2743, 728, F27, 22) (dual of [728, 685, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [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([727,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([727,22]) [i] based on
- 234 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 10 times 0, 1, 31 times 0, 1, 71 times 0, 1, 114 times 0) [i] based on linear OA(2747, 732, F27, 24) (dual of [732, 685, 25]-code), using
(55−24, 55, 739267)-Net in Base 27 — Upper bound on s
There is no (31, 55, 739268)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 5 308943 827083 008003 055795 507880 688599 840914 876535 389872 622565 351931 863476 371457 > 2755 [i]