Best Known (41−11, 41, s)-Nets in Base 3
(41−11, 41, 144)-Net over F3 — Constructive and digital
Digital (30, 41, 144)-net over F3, using
- 1 times m-reduction [i] based on digital (30, 42, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 14, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- trace code for nets [i] based on digital (2, 14, 48)-net over F27, using
(41−11, 41, 265)-Net over F3 — Digital
Digital (30, 41, 265)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(341, 265, F3, 11) (dual of [265, 224, 12]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 0, 0, 0) [i] based on linear OA(337, 253, F3, 11) (dual of [253, 216, 12]-code), using
- construction XX applied to C1 = C([112,121]), C2 = C([114,122]), C3 = C1 + C2 = C([114,121]), and C∩ = C1 ∩ C2 = C([112,122]) [i] based on
- linear OA(331, 242, F3, 10) (dual of [242, 211, 11]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {112,113,…,121}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(331, 242, F3, 9) (dual of [242, 211, 10]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {114,115,…,122}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(336, 242, F3, 11) (dual of [242, 206, 12]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {112,113,…,122}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(326, 242, F3, 8) (dual of [242, 216, 9]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {114,115,…,121}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(31, 6, F3, 1) (dual of [6, 5, 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, 5, F3, 0) (dual of [5, 5, 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([112,121]), C2 = C([114,122]), C3 = C1 + C2 = C([114,121]), and C∩ = C1 ∩ C2 = C([112,122]) [i] based on
- 8 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 0, 0, 0) [i] based on linear OA(337, 253, F3, 11) (dual of [253, 216, 12]-code), using
(41−11, 41, 8541)-Net in Base 3 — Upper bound on s
There is no (30, 41, 8542)-net in base 3, because
- 1 times m-reduction [i] would yield (30, 40, 8542)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 12 159338 937584 031189 > 340 [i]