Best Known (29, 39, s)-Nets in Base 3
(29, 39, 164)-Net over F3 — Constructive and digital
Digital (29, 39, 164)-net over F3, using
- base reduction for projective spaces (embedding PG(19,9) in PG(38,3)) for nets [i] based on digital (10, 20, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 82)-net over F81, using
(29, 39, 371)-Net over F3 — Digital
Digital (29, 39, 371)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(339, 371, F3, 2, 10) (dual of [(371, 2), 703, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(339, 742, F3, 10) (dual of [742, 703, 11]-code), using
- construction XX applied to C1 = C([360,367]), C2 = C([358,365]), C3 = C1 + C2 = C([360,365]), and C∩ = C1 ∩ C2 = C([358,367]) [i] based on
- linear OA(331, 728, F3, 8) (dual of [728, 697, 9]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {360,361,…,367}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(331, 728, F3, 8) (dual of [728, 697, 9]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {358,359,…,365}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(337, 728, F3, 10) (dual of [728, 691, 11]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {358,359,…,367}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(325, 728, F3, 6) (dual of [728, 703, 7]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {360,361,…,365}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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, 7, F3, 1) (dual of [7, 6, 2]-code) (see above)
- construction XX applied to C1 = C([360,367]), C2 = C([358,365]), C3 = C1 + C2 = C([360,365]), and C∩ = C1 ∩ C2 = C([358,367]) [i] based on
- OOA 2-folding [i] based on linear OA(339, 742, F3, 10) (dual of [742, 703, 11]-code), using
(29, 39, 6855)-Net in Base 3 — Upper bound on s
There is no (29, 39, 6856)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4 052725 449102 254865 > 339 [i]