Best Known (56−14, 56, s)-Nets in Base 3
(56−14, 56, 328)-Net over F3 — Constructive and digital
Digital (42, 56, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 14, 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
(56−14, 56, 396)-Net over F3 — Digital
Digital (42, 56, 396)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(356, 396, F3, 14) (dual of [396, 340, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(356, 741, F3, 14) (dual of [741, 685, 15]-code), using
- construction XX applied to C1 = C([352,364]), C2 = C([354,365]), C3 = C1 + C2 = C([354,364]), and C∩ = C1 ∩ C2 = C([352,365]) [i] based on
- linear OA(349, 728, F3, 13) (dual of [728, 679, 14]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {352,353,…,364}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(349, 728, F3, 12) (dual of [728, 679, 13]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {354,355,…,365}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(355, 728, F3, 14) (dual of [728, 673, 15]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {352,353,…,365}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(343, 728, F3, 11) (dual of [728, 685, 12]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {354,355,…,364}, and designed minimum distance d ≥ |I|+1 = 12 [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(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([352,364]), C2 = C([354,365]), C3 = C1 + C2 = C([354,364]), and C∩ = C1 ∩ C2 = C([352,365]) [i] based on
- discarding factors / shortening the dual code based on linear OA(356, 741, F3, 14) (dual of [741, 685, 15]-code), using
(56−14, 56, 11081)-Net in Base 3 — Upper bound on s
There is no (42, 56, 11082)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 523 466146 286135 981349 128265 > 356 [i]