Best Known (41, 56, s)-Nets in Base 3
(41, 56, 156)-Net over F3 — Constructive and digital
Digital (41, 56, 156)-net over F3, using
- 1 times m-reduction [i] based on digital (41, 57, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 19, 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, 19, 52)-net over F27, using
(41, 56, 272)-Net over F3 — Digital
Digital (41, 56, 272)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(356, 272, F3, 15) (dual of [272, 216, 16]-code), using
- 15 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 0, 0, 0, 1, 6 times 0) [i] based on linear OA(351, 252, F3, 15) (dual of [252, 201, 16]-code), using
- construction XX applied to C1 = C([241,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([241,13]) [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 = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(346, 242, F3, 14) (dual of [242, 196, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(351, 242, F3, 15) (dual of [242, 191, 16]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(341, 242, F3, 13) (dual of [242, 201, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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
- linear OA(30, 5, F3, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([241,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([241,13]) [i] based on
- 15 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 0, 0, 0, 1, 6 times 0) [i] based on linear OA(351, 252, F3, 15) (dual of [252, 201, 16]-code), using
(41, 56, 9471)-Net in Base 3 — Upper bound on s
There is no (41, 56, 9472)-net in base 3, because
- 1 times m-reduction [i] would yield (41, 55, 9472)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 174 564582 352761 370671 365121 > 355 [i]