Best Known (62, 62+13, s)-Nets in Base 8
(62, 62+13, 87382)-Net over F8 — Constructive and digital
Digital (62, 75, 87382)-net over F8, using
- 81 times duplication [i] based on digital (61, 74, 87382)-net over F8, using
- net defined by OOA [i] based on linear OOA(874, 87382, F8, 13, 13) (dual of [(87382, 13), 1135892, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(874, 524293, F8, 13) (dual of [524293, 524219, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 524294, F8, 13) (dual of [524294, 524220, 14]-code), using
- trace code [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
- trace code [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(874, 524294, F8, 13) (dual of [524294, 524220, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(874, 524293, F8, 13) (dual of [524293, 524219, 14]-code), using
- net defined by OOA [i] based on linear OOA(874, 87382, F8, 13, 13) (dual of [(87382, 13), 1135892, 14]-NRT-code), using
(62, 62+13, 524296)-Net over F8 — Digital
Digital (62, 75, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(875, 524296, F8, 13) (dual of [524296, 524221, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(874, 524294, F8, 13) (dual of [524294, 524220, 14]-code), using
- trace code [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
- trace code [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- linear OA(874, 524295, F8, 12) (dual of [524295, 524221, 13]-code), using Gilbert–Varšamov bound and bm = 874 > Vbs−1(k−1) = 40757 001873 443121 110418 312756 653790 108451 026759 978329 619837 091840 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(874, 524294, F8, 13) (dual of [524294, 524220, 14]-code), using
- construction X with Varšamov bound [i] based on
(62, 62+13, large)-Net in Base 8 — Upper bound on s
There is no (62, 75, large)-net in base 8, because
- 11 times m-reduction [i] would yield (62, 64, large)-net in base 8, but