Best Known (24, 24+25, s)-Nets in Base 64
(24, 24+25, 341)-Net over F64 — Constructive and digital
Digital (24, 49, 341)-net over F64, using
- net defined by OOA [i] based on linear OOA(6449, 341, F64, 25, 25) (dual of [(341, 25), 8476, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6449, 4093, F64, 25) (dual of [4093, 4044, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(6449, 4093, F64, 25) (dual of [4093, 4044, 26]-code), using
(24, 24+25, 407)-Net in Base 64 — Constructive
(24, 49, 407)-net in base 64, using
- base change [i] based on (17, 42, 407)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 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
- (4, 29, 257)-net in base 128, using
- 3 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
- base change [i] based on digital (0, 28, 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, 28, 257)-net over F256, using
- 3 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
- digital (1, 13, 150)-net over F128, using
- (u, u+v)-construction [i] based on
(24, 24+25, 1236)-Net over F64 — Digital
Digital (24, 49, 1236)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6449, 1236, F64, 3, 25) (dual of [(1236, 3), 3659, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6449, 1366, F64, 3, 25) (dual of [(1366, 3), 4049, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6449, 4098, F64, 25) (dual of [4098, 4049, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(24) ⊂ Ce(23) [i] based on
- OOA 3-folding [i] based on linear OA(6449, 4098, F64, 25) (dual of [4098, 4049, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(6449, 1366, F64, 3, 25) (dual of [(1366, 3), 4049, 26]-NRT-code), using
(24, 24+25, 1408442)-Net in Base 64 — Upper bound on s
There is no (24, 49, 1408443)-net in base 64, because
- 1 times m-reduction [i] would yield (24, 48, 1408443)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 497 326949 193874 547816 087683 560233 309462 976742 862790 345250 666454 688015 459312 384101 786620 > 6448 [i]