Best Known (28, 40, s)-Nets in Base 64
(28, 40, 43756)-Net over F64 — Constructive and digital
Digital (28, 40, 43756)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (22, 34, 43691)-net over F64, using
- net defined by OOA [i] based on linear OOA(6434, 43691, F64, 12, 12) (dual of [(43691, 12), 524258, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6434, 262146, F64, 12) (dual of [262146, 262112, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6434, 262147, F64, 12) (dual of [262147, 262113, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 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(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(6434, 262147, F64, 12) (dual of [262147, 262113, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6434, 262146, F64, 12) (dual of [262146, 262112, 13]-code), using
- net defined by OOA [i] based on linear OOA(6434, 43691, F64, 12, 12) (dual of [(43691, 12), 524258, 13]-NRT-code), using
- digital (0, 6, 65)-net over F64, using
(28, 40, 288144)-Net over F64 — Digital
Digital (28, 40, 288144)-net over F64, using
(28, 40, 349525)-Net in Base 64 — Constructive
(28, 40, 349525)-net in base 64, using
- net defined by OOA [i] based on OOA(6440, 349525, S64, 12, 12), using
- OA 6-folding and stacking [i] based on OA(6440, 2097150, S64, 12), using
- discarding factors based on OA(6440, 2097155, S64, 12), using
- discarding parts of the base [i] based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- discarding factors based on OA(6440, 2097155, S64, 12), using
- OA 6-folding and stacking [i] based on OA(6440, 2097150, S64, 12), using
(28, 40, 507245)-Net in Base 64
(28, 40, 507245)-net in base 64, using
- net defined by OOA [i] based on OOA(6440, 507245, S64, 15, 12), using
- OOA 2-folding and stacking with additional row [i] based on OOA(6440, 1014491, S64, 3, 12), using
- discarding parts of the base [i] based on linear OOA(12834, 1014491, F128, 3, 12) (dual of [(1014491, 3), 3043439, 13]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12834, 1014491, F128, 2, 12) (dual of [(1014491, 2), 2028948, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12834, 1048577, F128, 2, 12) (dual of [(1048577, 2), 2097120, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12834, 2097154, F128, 12) (dual of [2097154, 2097120, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(12834, 2097155, F128, 12) (dual of [2097155, 2097121, 13]-code), using
- OOA 2-folding [i] based on linear OA(12834, 2097154, F128, 12) (dual of [2097154, 2097120, 13]-code), using
- discarding factors / shortening the dual code based on linear OOA(12834, 1048577, F128, 2, 12) (dual of [(1048577, 2), 2097120, 13]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12834, 1014491, F128, 2, 12) (dual of [(1014491, 2), 2028948, 13]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(12834, 1014491, F128, 3, 12) (dual of [(1014491, 3), 3043439, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on OOA(6440, 1014491, S64, 3, 12), using
(28, 40, large)-Net in Base 64 — Upper bound on s
There is no (28, 40, large)-net in base 64, because
- 10 times m-reduction [i] would yield (28, 30, large)-net in base 64, but