Best Known (44−11, 44, s)-Nets in Base 3
(44−11, 44, 328)-Net over F3 — Constructive and digital
Digital (33, 44, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 11, 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
(44−11, 44, 387)-Net over F3 — Digital
Digital (33, 44, 387)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(344, 387, F3, 11) (dual of [387, 343, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(344, 741, F3, 11) (dual of [741, 697, 12]-code), using
- construction XX applied to C1 = C([355,364]), C2 = C([357,365]), C3 = C1 + C2 = C([357,364]), and C∩ = C1 ∩ C2 = C([355,365]) [i] based on
- 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 = {355,356,…,364}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(337, 728, F3, 9) (dual of [728, 691, 10]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {357,358,…,365}, and designed minimum distance d ≥ |I|+1 = 10 [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 = {355,356,…,365}, and designed minimum distance d ≥ |I|+1 = 12 [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 = {357,358,…,364}, and designed minimum distance d ≥ |I|+1 = 9 [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([355,364]), C2 = C([357,365]), C3 = C1 + C2 = C([357,364]), and C∩ = C1 ∩ C2 = C([355,365]) [i] based on
- discarding factors / shortening the dual code based on linear OA(344, 741, F3, 11) (dual of [741, 697, 12]-code), using
(44−11, 44, 16516)-Net in Base 3 — Upper bound on s
There is no (33, 44, 16517)-net in base 3, because
- 1 times m-reduction [i] would yield (33, 43, 16517)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 328 258543 533148 081179 > 343 [i]