Best Known (53−10, 53, s)-Nets in Base 8
(53−10, 53, 52433)-Net over F8 — Constructive and digital
Digital (43, 53, 52433)-net over F8, using
- net defined by OOA [i] based on linear OOA(853, 52433, F8, 10, 10) (dual of [(52433, 10), 524277, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(853, 262165, F8, 10) (dual of [262165, 262112, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(853, 262166, F8, 10) (dual of [262166, 262113, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(849, 262144, F8, 10) (dual of [262144, 262095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(831, 262144, F8, 6) (dual of [262144, 262113, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(84, 22, F8, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,8)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(853, 262166, F8, 10) (dual of [262166, 262113, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(853, 262165, F8, 10) (dual of [262165, 262112, 11]-code), using
(53−10, 53, 262166)-Net over F8 — Digital
Digital (43, 53, 262166)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(853, 262166, F8, 10) (dual of [262166, 262113, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(849, 262144, F8, 10) (dual of [262144, 262095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(831, 262144, F8, 6) (dual of [262144, 262113, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(84, 22, F8, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,8)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
(53−10, 53, large)-Net in Base 8 — Upper bound on s
There is no (43, 53, large)-net in base 8, because
- 8 times m-reduction [i] would yield (43, 45, large)-net in base 8, but