Best Known (70, 70+21, s)-Nets in Base 3
(70, 70+21, 400)-Net over F3 — Constructive and digital
Digital (70, 91, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (70, 92, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 23, 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, 23, 100)-net over F81, using
(70, 70+21, 705)-Net over F3 — Digital
Digital (70, 91, 705)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(391, 705, F3, 21) (dual of [705, 614, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(391, 758, F3, 21) (dual of [758, 667, 22]-code), using
- construction XX applied to C1 = C([724,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([724,16]) [i] based on
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,15}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(367, 728, F3, 17) (dual of [728, 661, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(385, 728, F3, 21) (dual of [728, 643, 22]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,16}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(361, 728, F3, 16) (dual of [728, 667, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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([724,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([724,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(391, 758, F3, 21) (dual of [758, 667, 22]-code), using
(70, 70+21, 44559)-Net in Base 3 — Upper bound on s
There is no (70, 91, 44560)-net in base 3, because
- 1 times m-reduction [i] would yield (70, 90, 44560)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8 727994 442795 517751 439080 671495 195846 604641 > 390 [i]