Best Known (45−23, 45, s)-Nets in Base 64
(45−23, 45, 372)-Net over F64 — Constructive and digital
Digital (22, 45, 372)-net over F64, using
- net defined by OOA [i] based on linear OOA(6445, 372, F64, 23, 23) (dual of [(372, 23), 8511, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6445, 4093, F64, 23) (dual of [4093, 4048, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6445, 4093, F64, 23) (dual of [4093, 4048, 24]-code), using
(45−23, 45, 407)-Net in Base 64 — Constructive
(22, 45, 407)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 14, 150)-net in base 64, using
- base change [i] based on digital (1, 12, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 12, 150)-net over F128, using
- (8, 31, 257)-net in base 64, using
- 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- (3, 14, 150)-net in base 64, using
(45−23, 45, 1229)-Net over F64 — Digital
Digital (22, 45, 1229)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6445, 1229, F64, 3, 23) (dual of [(1229, 3), 3642, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6445, 1366, F64, 3, 23) (dual of [(1366, 3), 4053, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 3-folding [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(6445, 1366, F64, 3, 23) (dual of [(1366, 3), 4053, 24]-NRT-code), using
(45−23, 45, 1307349)-Net in Base 64 — Upper bound on s
There is no (22, 45, 1307350)-net in base 64, because
- 1 times m-reduction [i] would yield (22, 44, 1307350)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 29 642840 559255 572161 251307 819048 339031 088544 125940 831446 745630 670533 073154 016036 > 6444 [i]