Best Known (38, 53, s)-Nets in Base 64
(38, 53, 37579)-Net over F64 — Constructive and digital
Digital (38, 53, 37579)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 130)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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, 7, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 3, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (28, 43, 37449)-net over F64, using
- net defined by OOA [i] based on linear OOA(6443, 37449, F64, 15, 15) (dual of [(37449, 15), 561692, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6443, 262144, F64, 15) (dual of [262144, 262101, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(6443, 262144, F64, 15) (dual of [262144, 262101, 16]-code), using
- net defined by OOA [i] based on linear OOA(6443, 37449, F64, 15, 15) (dual of [(37449, 15), 561692, 16]-NRT-code), using
- digital (3, 10, 130)-net over F64, using
(38, 53, 299594)-Net in Base 64 — Constructive
(38, 53, 299594)-net in base 64, using
- 641 times duplication [i] based on (37, 52, 299594)-net in base 64, using
- net defined by OOA [i] based on OOA(6452, 299594, S64, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(6452, 2097159, S64, 15), using
- discarding factors based on OA(6452, 2097160, S64, 15), using
- discarding parts of the base [i] based on linear OA(12844, 2097160, F128, 15) (dual of [2097160, 2097116, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding parts of the base [i] based on linear OA(12844, 2097160, F128, 15) (dual of [2097160, 2097116, 16]-code), using
- discarding factors based on OA(6452, 2097160, S64, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(6452, 2097159, S64, 15), using
- net defined by OOA [i] based on OOA(6452, 299594, S64, 15, 15), using
(38, 53, 660394)-Net over F64 — Digital
Digital (38, 53, 660394)-net over F64, using
(38, 53, large)-Net in Base 64 — Upper bound on s
There is no (38, 53, large)-net in base 64, because
- 13 times m-reduction [i] would yield (38, 40, large)-net in base 64, but