Best Known (46−23, 46, s)-Nets in Base 64
(46−23, 46, 372)-Net over F64 — Constructive and digital
Digital (23, 46, 372)-net over F64, using
- 641 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(6445, 372, F64, 23, 23) (dual of [(372, 23), 8511, 24]-NRT-code), using
(46−23, 46, 514)-Net in Base 64 — Constructive
(23, 46, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 15, 257)-net in base 64, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
- base change [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, 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 (see above)
- 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
- (4, 15, 257)-net in base 64, using
(46−23, 46, 1367)-Net over F64 — Digital
Digital (23, 46, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6446, 1367, F64, 3, 23) (dual of [(1367, 3), 4055, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6446, 4101, F64, 23) (dual of [4101, 4055, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6446, 4102, F64, 23) (dual of [4102, 4056, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(6445, 4097, F64, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6446, 4102, F64, 23) (dual of [4102, 4056, 24]-code), using
- OOA 3-folding [i] based on linear OA(6446, 4101, F64, 23) (dual of [4101, 4055, 24]-code), using
(46−23, 46, 1908053)-Net in Base 64 — Upper bound on s
There is no (23, 46, 1908054)-net in base 64, because
- 1 times m-reduction [i] would yield (23, 45, 1908054)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 1897 145144 872807 081463 657394 644365 502392 601726 027034 008463 594931 032013 209145 268548 > 6445 [i]