Best Known (112−20, 112, s)-Nets in Base 8
(112−20, 112, 26218)-Net over F8 — Constructive and digital
Digital (92, 112, 26218)-net over F8, using
- 82 times duplication [i] based on digital (90, 110, 26218)-net over F8, using
- net defined by OOA [i] based on linear OOA(8110, 26218, F8, 20, 20) (dual of [(26218, 20), 524250, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8110, 262180, F8, 20) (dual of [262180, 262070, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8110, 262181, F8, 20) (dual of [262181, 262071, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(8103, 262144, F8, 20) (dual of [262144, 262041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(873, 262144, F8, 14) (dual of [262144, 262071, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(8110, 262181, F8, 20) (dual of [262181, 262071, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8110, 262180, F8, 20) (dual of [262180, 262070, 21]-code), using
- net defined by OOA [i] based on linear OOA(8110, 26218, F8, 20, 20) (dual of [(26218, 20), 524250, 21]-NRT-code), using
(112−20, 112, 262185)-Net over F8 — Digital
Digital (92, 112, 262185)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8112, 262185, F8, 20) (dual of [262185, 262073, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8110, 262181, F8, 20) (dual of [262181, 262071, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(8103, 262144, F8, 20) (dual of [262144, 262041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(873, 262144, F8, 14) (dual of [262144, 262071, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(8110, 262183, F8, 19) (dual of [262183, 262073, 20]-code), using Gilbert–Varšamov bound and bm = 8110 > Vbs−1(k−1) = 8 710161 600991 020368 688753 382132 765152 720519 279504 588499 234014 466053 973854 027834 089049 412081 126492 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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(8110, 262181, F8, 20) (dual of [262181, 262071, 21]-code), using
- construction X with Varšamov bound [i] based on
(112−20, 112, large)-Net in Base 8 — Upper bound on s
There is no (92, 112, large)-net in base 8, because
- 18 times m-reduction [i] would yield (92, 94, large)-net in base 8, but