Best Known (79, 79+24, s)-Nets in Base 3
(79, 79+24, 400)-Net over F3 — Constructive and digital
Digital (79, 103, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (79, 104, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 26, 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, 26, 100)-net over F81, using
(79, 79+24, 718)-Net over F3 — Digital
Digital (79, 103, 718)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3103, 718, F3, 24) (dual of [718, 615, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3103, 758, F3, 24) (dual of [758, 655, 25]-code), using
- construction XX applied to C1 = C([724,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([724,19]) [i] based on
- linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,18}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(397, 728, F3, 24) (dual of [728, 631, 25]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,19}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(373, 728, F3, 19) (dual of [728, 655, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [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([724,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([724,19]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3103, 758, F3, 24) (dual of [758, 655, 25]-code), using
(79, 79+24, 32920)-Net in Base 3 — Upper bound on s
There is no (79, 103, 32921)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13 916150 708220 245274 782998 848971 286091 489875 407409 > 3103 [i]