Best Known (164, 196, s)-Nets in Base 3
(164, 196, 1480)-Net over F3 — Constructive and digital
Digital (164, 196, 1480)-net over F3, using
- t-expansion [i] based on digital (163, 196, 1480)-net over F3, 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
(164, 196, 9047)-Net over F3 — Digital
Digital (164, 196, 9047)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3196, 9047, F3, 2, 32) (dual of [(9047, 2), 17898, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3196, 9858, F3, 2, 32) (dual of [(9858, 2), 19520, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3196, 19716, F3, 32) (dual of [19716, 19520, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- linear OA(3190, 19683, F3, 32) (dual of [19683, 19493, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3196, 19716, F3, 32) (dual of [19716, 19520, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(3196, 9858, F3, 2, 32) (dual of [(9858, 2), 19520, 33]-NRT-code), using
(164, 196, 2378161)-Net in Base 3 — Upper bound on s
There is no (164, 196, 2378162)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3279 205181 152644 749460 752316 191517 451201 655481 811224 762400 808937 190897 926370 713256 359484 025505 > 3196 [i]