Best Known (28, 28+8, s)-Nets in Base 64
(28, 28+8, 2099199)-Net over F64 — Constructive and digital
Digital (28, 36, 2099199)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 2049)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, 2049, F64, 4, 4) (dual of [(2049, 4), 8189, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(647, 2049, F64, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(645, 4096, F64, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- appending kth column [i] based on linear OOA(647, 2049, F64, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(647, 2049, F64, 4, 4) (dual of [(2049, 4), 8189, 5]-NRT-code), using
- digital (21, 29, 2097150)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 2097150, F64, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6429, 8388600, F64, 8) (dual of [8388600, 8388571, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(6429, 8388600, F64, 8) (dual of [8388600, 8388571, 9]-code), using
- net defined by OOA [i] based on linear OOA(6429, 2097150, F64, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- digital (3, 7, 2049)-net over F64, using
(28, 28+8, 2105278)-Net in Base 64 — Constructive
(28, 36, 2105278)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 7, 8128)-net in base 64, using
- base change [i] based on digital (2, 6, 8128)-net over F128, using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- 1 times truncation [i] based on linear OA(1287, 16257, F128, 5) (dual of [16257, 16250, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(1286, 16256, F128, 4) (dual of [16256, 16250, 5]-code), using
- net defined by OOA [i] based on linear OOA(1286, 8128, F128, 4, 4) (dual of [(8128, 4), 32506, 5]-NRT-code), using
- base change [i] based on digital (2, 6, 8128)-net over F128, using
- digital (21, 29, 2097150)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 2097150, F64, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6429, 8388600, F64, 8) (dual of [8388600, 8388571, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(6429, 8388600, F64, 8) (dual of [8388600, 8388571, 9]-code), using
- net defined by OOA [i] based on linear OOA(6429, 2097150, F64, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- (3, 7, 8128)-net in base 64, using
(28, 28+8, large)-Net over F64 — Digital
Digital (28, 36, large)-net over F64, using
- 2 times m-reduction [i] based on digital (28, 38, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6438, large, F64, 10) (dual of [large, large−38, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 1 times code embedding in larger space [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6438, large, F64, 10) (dual of [large, large−38, 11]-code), using
(28, 28+8, large)-Net in Base 64 — Upper bound on s
There is no (28, 36, large)-net in base 64, because
- 6 times m-reduction [i] would yield (28, 30, large)-net in base 64, but