Best Known (32, 32+13, s)-Nets in Base 64
(32, 32+13, 43771)-Net over F64 — Constructive and digital
Digital (32, 45, 43771)-net over F64, using
- 641 times duplication [i] based on digital (31, 44, 43771)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (24, 37, 43691)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- digital (1, 7, 80)-net over F64, using
- (u, u+v)-construction [i] based on
(32, 32+13, 349526)-Net in Base 64 — Constructive
(32, 45, 349526)-net in base 64, using
- net defined by OOA [i] based on OOA(6445, 349526, S64, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6445, 2097157, S64, 13), using
- discarding factors based on OA(6445, 2097160, S64, 13), using
- discarding parts of the base [i] based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- discarding factors based on OA(6445, 2097160, S64, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6445, 2097157, S64, 13), using
(32, 32+13, 497967)-Net over F64 — Digital
Digital (32, 45, 497967)-net over F64, using
(32, 32+13, 524289)-Net in Base 64
(32, 45, 524289)-net in base 64, using
- net defined by OOA [i] based on OOA(6445, 524289, S64, 15, 13), using
- OOA 2-folding and stacking with additional row [i] based on OOA(6445, 1048579, S64, 3, 13), using
- discarding factors based on OOA(6445, 1048580, S64, 3, 13), using
- discarding parts of the base [i] based on linear OOA(12838, 1048580, F128, 3, 13) (dual of [(1048580, 3), 3145702, 14]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12838, 1048580, F128, 2, 13) (dual of [(1048580, 2), 2097122, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- OOA 2-folding [i] based on linear OA(12838, 2097160, F128, 13) (dual of [2097160, 2097122, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12838, 1048580, F128, 2, 13) (dual of [(1048580, 2), 2097122, 14]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(12838, 1048580, F128, 3, 13) (dual of [(1048580, 3), 3145702, 14]-NRT-code), using
- discarding factors based on OOA(6445, 1048580, S64, 3, 13), using
- OOA 2-folding and stacking with additional row [i] based on OOA(6445, 1048579, S64, 3, 13), using
(32, 32+13, large)-Net in Base 64 — Upper bound on s
There is no (32, 45, large)-net in base 64, because
- 11 times m-reduction [i] would yield (32, 34, large)-net in base 64, but