Best Known (49−26, 49, s)-Nets in Base 64
(49−26, 49, 281)-Net over F64 — Constructive and digital
Digital (23, 49, 281)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 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, 33, 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, 16, 104)-net over F64, using
(49−26, 49, 337)-Net in Base 64 — Constructive
(23, 49, 337)-net in base 64, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 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
- (9, 35, 257)-net in base 64, using
- 1 times m-reduction [i] based on (9, 36, 257)-net in base 64, using
- base change [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
- 1 times m-reduction [i] based on (9, 36, 257)-net in base 64, using
- digital (1, 14, 80)-net over F64, using
(49−26, 49, 589)-Net over F64 — Digital
Digital (23, 49, 589)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6449, 589, F64, 26) (dual of [589, 540, 27]-code), using
- construction XX applied to C1 = C([51,75]), C2 = C([50,74]), C3 = C1 + C2 = C([51,74]), and C∩ = C1 ∩ C2 = C([50,75]) [i] based on
- linear OA(6447, 585, F64, 25) (dual of [585, 538, 26]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {51,52,…,75}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(6447, 585, F64, 25) (dual of [585, 538, 26]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {50,51,…,74}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(6449, 585, F64, 26) (dual of [585, 536, 27]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {50,51,…,75}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(6445, 585, F64, 24) (dual of [585, 540, 25]-code), using the BCH-code C(I) with length 585 | 642−1, defining interval I = {51,52,…,74}, and designed minimum distance d ≥ |I|+1 = 25 [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([51,75]), C2 = C([50,74]), C3 = C1 + C2 = C([51,74]), and C∩ = C1 ∩ C2 = C([50,75]) [i] based on
(49−26, 49, 578056)-Net in Base 64 — Upper bound on s
There is no (23, 49, 578057)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 31829 226404 956351 711616 759649 602372 867736 089194 819778 929940 005783 628813 119047 069220 641280 > 6449 [i]