Best Known (37, 52, s)-Nets in Base 64
(37, 52, 37530)-Net over F64 — Constructive and digital
Digital (37, 52, 37530)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (29, 44, 37450)-net over F64, using
- net defined by OOA [i] based on linear OOA(6444, 37450, F64, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6444, 262151, F64, 15) (dual of [262151, 262107, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 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([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6444, 262151, F64, 15) (dual of [262151, 262107, 16]-code), using
- net defined by OOA [i] based on linear OOA(6444, 37450, F64, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
- digital (1, 8, 80)-net over F64, using
(37, 52, 299594)-Net in Base 64 — Constructive
(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
(37, 52, 490673)-Net over F64 — Digital
Digital (37, 52, 490673)-net over F64, using
(37, 52, large)-Net in Base 64 — Upper bound on s
There is no (37, 52, large)-net in base 64, because
- 13 times m-reduction [i] would yield (37, 39, large)-net in base 64, but