Best Known (146−30, 146, s)-Nets in Base 3
(146−30, 146, 640)-Net over F3 — Constructive and digital
Digital (116, 146, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (116, 148, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 37, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 37, 160)-net over F81, using
(146−30, 146, 1644)-Net over F3 — Digital
Digital (116, 146, 1644)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3146, 1644, F3, 30) (dual of [1644, 1498, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3146, 2209, F3, 30) (dual of [2209, 2063, 31]-code), using
- construction XX applied to Ce(30) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3120, 2187, F3, 26) (dual of [2187, 2067, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(30) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3146, 2209, F3, 30) (dual of [2209, 2063, 31]-code), using
(146−30, 146, 141472)-Net in Base 3 — Upper bound on s
There is no (116, 146, 141473)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4567 950500 179650 704642 685366 923861 336868 500129 992734 291648 219032 616171 > 3146 [i]