Best Known (65, 65+21, s)-Nets in Base 3
(65, 65+21, 328)-Net over F3 — Constructive and digital
Digital (65, 86, 328)-net over F3, using
- 32 times duplication [i] based on digital (63, 84, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 21, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 21, 82)-net over F81, using
(65, 65+21, 524)-Net over F3 — Digital
Digital (65, 86, 524)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(386, 524, F3, 21) (dual of [524, 438, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(386, 743, F3, 21) (dual of [743, 657, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(385, 730, F3, 21) (dual of [730, 645, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(373, 730, F3, 19) (dual of [730, 657, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(386, 743, F3, 21) (dual of [743, 657, 22]-code), using
(65, 65+21, 25722)-Net in Base 3 — Upper bound on s
There is no (65, 86, 25723)-net in base 3, because
- 1 times m-reduction [i] would yield (65, 85, 25723)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 35921 667604 756416 600451 510419 754290 312261 > 385 [i]