Best Known (74, 74+29, s)-Nets in Base 9
(74, 74+29, 740)-Net over F9 — Constructive and digital
Digital (74, 103, 740)-net over F9, using
- 13 times m-reduction [i] based on digital (74, 116, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 58, 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
- trace code for nets [i] based on digital (16, 58, 370)-net over F81, using
(74, 74+29, 5483)-Net over F9 — Digital
Digital (74, 103, 5483)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9103, 5483, F9, 29) (dual of [5483, 5380, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(9103, 6571, F9, 29) (dual of [6571, 6468, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(9101, 6561, F9, 29) (dual of [6561, 6460, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(993, 6561, F9, 26) (dual of [6561, 6468, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(92, 10, F9, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,9)), using
- extended Reed–Solomon code RSe(8,9) [i]
- Hamming code H(2,9) [i]
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(9103, 6571, F9, 29) (dual of [6571, 6468, 30]-code), using
(74, 74+29, 6771821)-Net in Base 9 — Upper bound on s
There is no (74, 103, 6771822)-net in base 9, because
- 1 times m-reduction [i] would yield (74, 102, 6771822)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 21 514752 227078 267135 297627 547388 794931 971391 281075 142831 212312 008995 658960 484894 674043 156889 319649 > 9102 [i]