Best Known (35−17, 35, s)-Nets in Base 64
(35−17, 35, 512)-Net over F64 — Constructive and digital
Digital (18, 35, 512)-net over F64, using
- 642 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
(35−17, 35, 515)-Net in Base 64 — Constructive
(18, 35, 515)-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
- (7, 24, 258)-net in base 64, using
- base change [i] based on digital (1, 18, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 18, 258)-net over F256, using
- (3, 11, 257)-net in base 64, using
(35−17, 35, 1730)-Net over F64 — Digital
Digital (18, 35, 1730)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6435, 1730, F64, 2, 17) (dual of [(1730, 2), 3425, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6435, 2052, F64, 2, 17) (dual of [(2052, 2), 4069, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6435, 4104, F64, 17) (dual of [4104, 4069, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(6433, 4096, F64, 17) (dual of [4096, 4063, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(6435, 4104, F64, 17) (dual of [4104, 4069, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(6435, 2052, F64, 2, 17) (dual of [(2052, 2), 4069, 18]-NRT-code), using
(35−17, 35, 2835396)-Net in Base 64 — Upper bound on s
There is no (18, 35, 2835397)-net in base 64, because
- 1 times m-reduction [i] would yield (18, 34, 2835397)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 25 711020 856547 877009 162981 964585 329978 535043 774405 427321 903448 > 6434 [i]