Best Known (65, 90, s)-Nets in Base 9
(65, 90, 740)-Net over F9 — Constructive and digital
Digital (65, 90, 740)-net over F9, using
- 8 times m-reduction [i] based on digital (65, 98, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 49, 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, 49, 370)-net over F81, using
(65, 90, 5793)-Net over F9 — Digital
Digital (65, 90, 5793)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(990, 5793, F9, 25) (dual of [5793, 5703, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(990, 6571, F9, 25) (dual of [6571, 6481, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(989, 6562, F9, 25) (dual of [6562, 6473, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(981, 6562, F9, 23) (dual of [6562, 6481, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [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 C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(990, 6571, F9, 25) (dual of [6571, 6481, 26]-code), using
(65, 90, 7898955)-Net in Base 9 — Upper bound on s
There is no (65, 90, 7898956)-net in base 9, because
- 1 times m-reduction [i] would yield (65, 89, 7898956)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 8 464159 411272 468033 591951 458494 048391 313200 409580 110265 685972 525740 540193 359555 945345 > 989 [i]