Best Known (55−15, 55, s)-Nets in Base 3
(55−15, 55, 156)-Net over F3 — Constructive and digital
Digital (40, 55, 156)-net over F3, using
- 31 times duplication [i] based on digital (39, 54, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 18, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- trace code for nets [i] based on digital (3, 18, 52)-net over F27, using
(55−15, 55, 261)-Net over F3 — Digital
Digital (40, 55, 261)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(355, 261, F3, 15) (dual of [261, 206, 16]-code), using
- construction XX applied to C1 = C([239,10]), C2 = C([1,12]), C3 = C1 + C2 = C([1,10]), and C∩ = C1 ∩ C2 = C([239,12]) [i] based on
- 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 = {−3,−2,…,10}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(340, 242, F3, 12) (dual of [242, 202, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(351, 242, F3, 16) (dual of [242, 191, 17]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−3,−2,…,12}, and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(335, 242, F3, 10) (dual of [242, 207, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [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
- construction XX applied to C1 = C([239,10]), C2 = C([1,12]), C3 = C1 + C2 = C([1,10]), and C∩ = C1 ∩ C2 = C([239,12]) [i] based on
(55−15, 55, 8094)-Net in Base 3 — Upper bound on s
There is no (40, 55, 8095)-net in base 3, because
- 1 times m-reduction [i] would yield (40, 54, 8095)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 58 175036 670105 392009 729139 > 354 [i]