Best Known (21, 21+12, s)-Nets in Base 64
(21, 21+12, 1618)-Net over F64 — Constructive and digital
Digital (21, 33, 1618)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 1618, F64, 12, 12) (dual of [(1618, 12), 19383, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6433, 9708, F64, 12) (dual of [9708, 9675, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6433, 9709, F64, 12) (dual of [9709, 9676, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6433, 9708, F64, 12) (dual of [9708, 9675, 13]-code), using
(21, 21+12, 10923)-Net in Base 64 — Constructive
(21, 33, 10923)-net in base 64, using
- 641 times duplication [i] based on (20, 32, 10923)-net in base 64, using
- base change [i] based on digital (12, 24, 10923)-net over F256, using
- 2561 times duplication [i] based on digital (11, 23, 10923)-net over F256, using
- net defined by OOA [i] based on linear OOA(25623, 10923, F256, 12, 12) (dual of [(10923, 12), 131053, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(25623, 65538, F256, 12) (dual of [65538, 65515, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(25623, 65538, F256, 12) (dual of [65538, 65515, 13]-code), using
- net defined by OOA [i] based on linear OOA(25623, 10923, F256, 12, 12) (dual of [(10923, 12), 131053, 13]-NRT-code), using
- 2561 times duplication [i] based on digital (11, 23, 10923)-net over F256, using
- base change [i] based on digital (12, 24, 10923)-net over F256, using
(21, 21+12, 20433)-Net over F64 — Digital
Digital (21, 33, 20433)-net over F64, using
(21, 21+12, 21847)-Net in Base 64
(21, 33, 21847)-net in base 64, using
- 641 times duplication [i] based on (20, 32, 21847)-net in base 64, using
- base change [i] based on digital (12, 24, 21847)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25624, 21847, F256, 3, 12) (dual of [(21847, 3), 65517, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25624, 65541, F256, 12) (dual of [65541, 65517, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OOA 3-folding [i] based on linear OA(25624, 65541, F256, 12) (dual of [65541, 65517, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25624, 21847, F256, 3, 12) (dual of [(21847, 3), 65517, 13]-NRT-code), using
- base change [i] based on digital (12, 24, 21847)-net over F256, using
(21, 21+12, large)-Net in Base 64 — Upper bound on s
There is no (21, 33, large)-net in base 64, because
- 10 times m-reduction [i] would yield (21, 23, large)-net in base 64, but