Best Known (24, 24+10, s)-Nets in Base 64
(24, 24+10, 52509)-Net over F64 — Constructive and digital
Digital (24, 34, 52509)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 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 (18, 28, 52429)-net over F64, using
- net defined by OOA [i] based on linear OOA(6428, 52429, F64, 10, 10) (dual of [(52429, 10), 524262, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6428, 262145, F64, 10) (dual of [262145, 262117, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6428, 262147, F64, 10) (dual of [262147, 262119, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6425, 262144, F64, 9) (dual of [262144, 262119, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(6428, 262147, F64, 10) (dual of [262147, 262119, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6428, 262145, F64, 10) (dual of [262145, 262117, 11]-code), using
- net defined by OOA [i] based on linear OOA(6428, 52429, F64, 10, 10) (dual of [(52429, 10), 524262, 11]-NRT-code), using
- digital (1, 6, 80)-net over F64, using
(24, 24+10, 419431)-Net in Base 64 — Constructive
(24, 34, 419431)-net in base 64, using
- 641 times duplication [i] based on (23, 33, 419431)-net in base 64, using
- net defined by OOA [i] based on OOA(6433, 419431, S64, 10, 10), using
- OA 5-folding and stacking [i] based on OA(6433, 2097155, S64, 10), using
- discarding parts of the base [i] based on linear OA(12828, 2097155, F128, 10) (dual of [2097155, 2097127, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding parts of the base [i] based on linear OA(12828, 2097155, F128, 10) (dual of [2097155, 2097127, 11]-code), using
- OA 5-folding and stacking [i] based on OA(6433, 2097155, S64, 10), using
- net defined by OOA [i] based on OOA(6433, 419431, S64, 10, 10), using
(24, 24+10, 438290)-Net over F64 — Digital
Digital (24, 34, 438290)-net over F64, using
(24, 24+10, 524289)-Net in Base 64
(24, 34, 524289)-net in base 64, using
- net defined by OOA [i] based on OOA(6434, 524289, S64, 15, 10), using
- OOA 2-folding and stacking with additional row [i] based on OOA(6434, 1048579, S64, 3, 10), using
- discarding parts of the base [i] based on linear OOA(12829, 1048579, F128, 3, 10) (dual of [(1048579, 3), 3145708, 11]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12829, 1048579, F128, 2, 10) (dual of [(1048579, 2), 2097129, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12829, 2097158, F128, 10) (dual of [2097158, 2097129, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12829, 2097159, F128, 10) (dual of [2097159, 2097130, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(12829, 2097159, F128, 10) (dual of [2097159, 2097130, 11]-code), using
- OOA 2-folding [i] based on linear OA(12829, 2097158, F128, 10) (dual of [2097158, 2097129, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12829, 1048579, F128, 2, 10) (dual of [(1048579, 2), 2097129, 11]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(12829, 1048579, F128, 3, 10) (dual of [(1048579, 3), 3145708, 11]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on OOA(6434, 1048579, S64, 3, 10), using
(24, 24+10, large)-Net in Base 64 — Upper bound on s
There is no (24, 34, large)-net in base 64, because
- 8 times m-reduction [i] would yield (24, 26, large)-net in base 64, but