Best Known (64, 64+18, s)-Nets in Base 64
(64, 64+18, 932197)-Net over F64 — Constructive and digital
Digital (64, 82, 932197)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 130)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (0, 9, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 4, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (51, 69, 932067)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- digital (4, 13, 130)-net over F64, using
(64, 64+18, 932324)-Net in Base 64 — Constructive
(64, 82, 932324)-net in base 64, using
- 641 times duplication [i] based on (63, 81, 932324)-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
- digital (51, 69, 932067)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- (3, 12, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(64, 64+18, large)-Net over F64 — Digital
Digital (64, 82, large)-net over F64, using
- t-expansion [i] based on digital (63, 82, large)-net over F64, using
- 2 times m-reduction [i] based on digital (63, 84, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6484, large, F64, 21) (dual of [large, large−84, 22]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6481, large, F64, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 3 times code embedding in larger space [i] based on linear OA(6481, large, F64, 21) (dual of [large, large−81, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6484, large, F64, 21) (dual of [large, large−84, 22]-code), using
- 2 times m-reduction [i] based on digital (63, 84, large)-net over F64, using
(64, 64+18, large)-Net in Base 64 — Upper bound on s
There is no (64, 82, large)-net in base 64, because
- 16 times m-reduction [i] would yield (64, 66, large)-net in base 64, but