Best Known (38−19, 38, s)-Nets in Base 64
(38−19, 38, 455)-Net over F64 — Constructive and digital
Digital (19, 38, 455)-net over F64, using
- 641 times duplication [i] based on digital (18, 37, 455)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 455, F64, 19, 19) (dual of [(455, 19), 8608, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using
- net defined by OOA [i] based on linear OOA(6437, 455, F64, 19, 19) (dual of [(455, 19), 8608, 20]-NRT-code), using
(38−19, 38, 514)-Net in Base 64 — Constructive
(19, 38, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 12, 257)-net in base 64, using
- base change [i] based on digital (0, 9, 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, 9, 257)-net over F256, using
- (7, 26, 257)-net in base 64, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 21, 257)-net over F256, using
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- (3, 12, 257)-net in base 64, using
(38−19, 38, 1367)-Net over F64 — Digital
Digital (19, 38, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6438, 1367, F64, 3, 19) (dual of [(1367, 3), 4063, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6438, 4101, F64, 19) (dual of [4101, 4063, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6438, 4102, F64, 19) (dual of [4102, 4064, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6438, 4102, F64, 19) (dual of [4102, 4064, 20]-code), using
- OOA 3-folding [i] based on linear OA(6438, 4101, F64, 19) (dual of [4101, 4063, 20]-code), using
(38−19, 38, 1753139)-Net in Base 64 — Upper bound on s
There is no (19, 38, 1753140)-net in base 64, because
- 1 times m-reduction [i] would yield (19, 37, 1753140)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 6 740009 955614 830917 637718 330647 523771 813720 736141 389727 624598 522210 > 6437 [i]