Best Known (56−10, 56, s)-Nets in Base 8
(56−10, 56, 104858)-Net over F8 — Constructive and digital
Digital (46, 56, 104858)-net over F8, using
- net defined by OOA [i] based on linear OOA(856, 104858, F8, 10, 10) (dual of [(104858, 10), 1048524, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(856, 524290, F8, 10) (dual of [524290, 524234, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 524294, F8, 10) (dual of [524294, 524238, 11]-code), using
- trace code [i] 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
- trace code [i] based on linear OA(6428, 262147, F64, 10) (dual of [262147, 262119, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 524294, F8, 10) (dual of [524294, 524238, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(856, 524290, F8, 10) (dual of [524290, 524234, 11]-code), using
(56−10, 56, 524294)-Net over F8 — Digital
Digital (46, 56, 524294)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(856, 524294, F8, 10) (dual of [524294, 524238, 11]-code), using
- trace code [i] 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
- trace code [i] based on linear OA(6428, 262147, F64, 10) (dual of [262147, 262119, 11]-code), using
(56−10, 56, large)-Net in Base 8 — Upper bound on s
There is no (46, 56, large)-net in base 8, because
- 8 times m-reduction [i] would yield (46, 48, large)-net in base 8, but