Best Known (18, 29, s)-Nets in Base 9
(18, 29, 232)-Net over F9 — Constructive and digital
Digital (18, 29, 232)-net over F9, using
- 3 times m-reduction [i] based on digital (18, 32, 232)-net over F9, using
- trace code for nets [i] based on digital (2, 16, 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
- trace code for nets [i] based on digital (2, 16, 116)-net over F81, using
(18, 29, 478)-Net over F9 — Digital
Digital (18, 29, 478)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(929, 478, F9, 11) (dual of [478, 449, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(929, 735, F9, 11) (dual of [735, 706, 12]-code), using
- construction XX applied to C1 = C([82,91]), C2 = C([84,92]), C3 = C1 + C2 = C([84,91]), and C∩ = C1 ∩ C2 = C([82,92]) [i] based on
- linear OA(925, 728, F9, 10) (dual of [728, 703, 11]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {82,83,…,91}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(925, 728, F9, 9) (dual of [728, 703, 10]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {84,85,…,92}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(928, 728, F9, 11) (dual of [728, 700, 12]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {82,83,…,92}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(922, 728, F9, 8) (dual of [728, 706, 9]-code), using the primitive BCH-code C(I) with length 728 = 93−1, defining interval I = {84,85,…,91}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(91, 4, F9, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- Reed–Solomon code RS(8,9) [i]
- discarding factors / shortening the dual code based on linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([82,91]), C2 = C([84,92]), C3 = C1 + C2 = C([84,91]), and C∩ = C1 ∩ C2 = C([82,92]) [i] based on
- discarding factors / shortening the dual code based on linear OA(929, 735, F9, 11) (dual of [735, 706, 12]-code), using
(18, 29, 71860)-Net in Base 9 — Upper bound on s
There is no (18, 29, 71861)-net in base 9, because
- 1 times m-reduction [i] would yield (18, 28, 71861)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 523 376586 152445 250739 007945 > 928 [i]