Best Known (61, 61+18, s)-Nets in Base 3
(61, 61+18, 400)-Net over F3 — Constructive and digital
Digital (61, 79, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (61, 80, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
(61, 61+18, 706)-Net over F3 — Digital
Digital (61, 79, 706)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(379, 706, F3, 18) (dual of [706, 627, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(379, 758, F3, 18) (dual of [758, 679, 19]-code), using
- construction XX applied to C1 = C([724,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([724,13]) [i] based on
- linear OA(367, 728, F3, 17) (dual of [728, 661, 18]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,12}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(355, 728, F3, 14) (dual of [728, 673, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(373, 728, F3, 18) (dual of [728, 655, 19]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,13}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(349, 728, F3, 13) (dual of [728, 679, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(36, 24, F3, 3) (dual of [24, 18, 4]-code or 24-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), 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([724,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([724,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(379, 758, F3, 18) (dual of [758, 679, 19]-code), using
(61, 61+18, 31964)-Net in Base 3 — Upper bound on s
There is no (61, 79, 31965)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 49 274068 730862 218989 134500 224082 442907 > 379 [i]