Best Known (26, 53, s)-Nets in Base 64
(26, 53, 315)-Net over F64 — Constructive and digital
Digital (26, 53, 315)-net over F64, using
- net defined by OOA [i] based on linear OOA(6453, 315, F64, 27, 27) (dual of [(315, 27), 8452, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(6453, 4096, F64, 27) (dual of [4096, 4043, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- OOA 13-folding and stacking with additional row [i] based on linear OA(6453, 4096, F64, 27) (dual of [4096, 4043, 28]-code), using
(26, 53, 407)-Net in Base 64 — Constructive
(26, 53, 407)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 17, 150)-net in base 64, using
- 4 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
- base change [i] based on digital (1, 18, 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, 18, 150)-net over F128, using
- 4 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
- (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
- (4, 17, 150)-net in base 64, using
(26, 53, 1253)-Net over F64 — Digital
Digital (26, 53, 1253)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6453, 1253, F64, 3, 27) (dual of [(1253, 3), 3706, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6453, 1366, F64, 3, 27) (dual of [(1366, 3), 4045, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6453, 4098, F64, 27) (dual of [4098, 4045, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(6453, 4096, F64, 27) (dual of [4096, 4043, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(6451, 4096, F64, 26) (dual of [4096, 4045, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- OOA 3-folding [i] based on linear OA(6453, 4098, F64, 27) (dual of [4098, 4045, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(6453, 1366, F64, 3, 27) (dual of [(1366, 3), 4045, 28]-NRT-code), using
(26, 53, 1509328)-Net in Base 64 — Upper bound on s
There is no (26, 53, 1509329)-net in base 64, because
- 1 times m-reduction [i] would yield (26, 52, 1509329)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 8343 743749 272849 371502 490070 428814 104466 597542 501444 413797 641691 343714 461193 611404 094266 749712 > 6452 [i]