Best Known (17, 17+17, s)-Nets in Base 64
(17, 17+17, 512)-Net over F64 — Constructive and digital
Digital (17, 34, 512)-net over F64, using
- 641 times duplication [i] based on digital (16, 33, 512)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 512, F64, 17, 17) (dual of [(512, 17), 8671, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on 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]
- OOA 8-folding and stacking with additional row [i] based on linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using
- net defined by OOA [i] based on linear OOA(6433, 512, F64, 17, 17) (dual of [(512, 17), 8671, 18]-NRT-code), using
(17, 17+17, 514)-Net in Base 64 — Constructive
(17, 34, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 11, 257)-net in base 64, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
- (6, 23, 257)-net in base 64, using
- 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- base change [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- (3, 11, 257)-net in base 64, using
(17, 17+17, 1367)-Net over F64 — Digital
Digital (17, 34, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6434, 1367, F64, 3, 17) (dual of [(1367, 3), 4067, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6434, 4101, F64, 17) (dual of [4101, 4067, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6434, 4102, F64, 17) (dual of [4102, 4068, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- 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(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6434, 4102, F64, 17) (dual of [4102, 4068, 18]-code), using
- OOA 3-folding [i] based on linear OA(6434, 4101, F64, 17) (dual of [4101, 4067, 18]-code), using
(17, 17+17, 1685935)-Net in Base 64 — Upper bound on s
There is no (17, 34, 1685936)-net in base 64, because
- 1 times m-reduction [i] would yield (17, 33, 1685936)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 401735 324689 581869 497779 169109 772219 853771 463969 678979 001735 > 6433 [i]