Best Known (26, 26+30, s)-Nets in Base 64
(26, 26+30, 281)-Net over F64 — Constructive and digital
Digital (26, 56, 281)-net over F64, using
- 2 times m-reduction [i] based on digital (26, 58, 281)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 19, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- digital (7, 39, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- digital (3, 19, 104)-net over F64, using
- (u, u+v)-construction [i] based on
(26, 26+30, 337)-Net in Base 64 — Constructive
(26, 56, 337)-net in base 64, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- (10, 40, 257)-net in base 64, using
- base change [i] based on digital (0, 30, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 30, 257)-net over F256, using
- digital (1, 16, 80)-net over F64, using
(26, 26+30, 589)-Net over F64 — Digital
Digital (26, 56, 589)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6456, 589, F64, 30) (dual of [589, 533, 31]-code), using
- construction XX applied to C1 = C([19,47]), C2 = C([18,46]), C3 = C1 + C2 = C([19,46]), and C∩ = C1 ∩ C2 = C([18,47]) [i] based on
- linear OA(6454, 585, F64, 29) (dual of [585, 531, 30]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {19,20,…,47}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(6454, 585, F64, 29) (dual of [585, 531, 30]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {18,19,…,46}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(6456, 585, F64, 30) (dual of [585, 529, 31]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {18,19,…,47}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(6452, 585, F64, 28) (dual of [585, 533, 29]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {19,20,…,46}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([19,47]), C2 = C([18,46]), C3 = C1 + C2 = C([19,46]), and C∩ = C1 ∩ C2 = C([18,47]) [i] based on
(26, 26+30, 564277)-Net in Base 64 — Upper bound on s
There is no (26, 56, 564278)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 139987 406388 991827 501216 753943 788667 732233 037492 042862 726077 099295 177944 401286 109173 252812 340508 086264 > 6456 [i]