Best Known (62, 62+24, s)-Nets in Base 9
(62, 62+24, 740)-Net over F9 — Constructive and digital
Digital (62, 86, 740)-net over F9, using
- 6 times m-reduction [i] based on digital (62, 92, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 46, 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, 46, 370)-net over F81, using
(62, 62+24, 5490)-Net over F9 — Digital
Digital (62, 86, 5490)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(986, 5490, F9, 24) (dual of [5490, 5404, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(986, 6570, F9, 24) (dual of [6570, 6484, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(985, 6561, F9, 24) (dual of [6561, 6476, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(977, 6561, F9, 22) (dual of [6561, 6484, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(986, 6570, F9, 24) (dual of [6570, 6484, 25]-code), using
(62, 62+24, 4560460)-Net in Base 9 — Upper bound on s
There is no (62, 86, 4560461)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 11610 631251 686892 200873 303560 442557 409193 528881 624694 946255 324301 142900 397094 908513 > 986 [i]