Best Known (57−10, 57, s)-Nets in Base 8
(57−10, 57, 419431)-Net over F8 — Constructive and digital
Digital (47, 57, 419431)-net over F8, using
- net defined by OOA [i] based on linear OOA(857, 419431, F8, 10, 10) (dual of [(419431, 10), 4194253, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(857, 2097155, F8, 10) (dual of [2097155, 2097098, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(857, 2097159, F8, 10) (dual of [2097159, 2097102, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(850, 2097152, F8, 9) (dual of [2097152, 2097102, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(857, 2097159, F8, 10) (dual of [2097159, 2097102, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(857, 2097155, F8, 10) (dual of [2097155, 2097098, 11]-code), using
(57−10, 57, 1127769)-Net over F8 — Digital
Digital (47, 57, 1127769)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(857, 1127769, F8, 10) (dual of [1127769, 1127712, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(857, 2097152, F8, 10) (dual of [2097152, 2097095, 11]-code), using
(57−10, 57, large)-Net in Base 8 — Upper bound on s
There is no (47, 57, large)-net in base 8, because
- 8 times m-reduction [i] would yield (47, 49, large)-net in base 8, but