Best Known (54−14, 54, s)-Nets in Base 3
(54−14, 54, 192)-Net over F3 — Constructive and digital
Digital (40, 54, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 18, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
(54−14, 54, 292)-Net over F3 — Digital
Digital (40, 54, 292)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(354, 292, F3, 14) (dual of [292, 238, 15]-code), using
- 32 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0) [i] based on linear OA(347, 253, F3, 14) (dual of [253, 206, 15]-code), using
- construction XX applied to C1 = C([109,121]), C2 = C([111,122]), C3 = C1 + C2 = C([111,121]), and C∩ = C1 ∩ C2 = C([109,122]) [i] based on
- linear OA(341, 242, F3, 13) (dual of [242, 201, 14]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {109,110,…,121}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(341, 242, F3, 12) (dual of [242, 201, 13]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {111,112,…,122}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(346, 242, F3, 14) (dual of [242, 196, 15]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {109,110,…,122}, and designed minimum distance d ≥ |I|+1 = 15 [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 = {111,112,…,121}, and designed minimum distance d ≥ |I|+1 = 12 [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([109,121]), C2 = C([111,122]), C3 = C1 + C2 = C([111,121]), and C∩ = C1 ∩ C2 = C([109,122]) [i] based on
- 32 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0) [i] based on linear OA(347, 253, F3, 14) (dual of [253, 206, 15]-code), using
(54−14, 54, 8094)-Net in Base 3 — Upper bound on s
There is no (40, 54, 8095)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 58 175036 670105 392009 729139 > 354 [i]