Best Known (91, 91+20, s)-Nets in Base 8
(91, 91+20, 26218)-Net over F8 — Constructive and digital
Digital (91, 111, 26218)-net over F8, using
- 81 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
(91, 91+20, 262183)-Net over F8 — Digital
Digital (91, 111, 262183)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8111, 262183, F8, 20) (dual of [262183, 262072, 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, 262182, F8, 19) (dual of [262182, 262072, 20]-code), using Gilbert–Varšamov bound and bm = 8110 > Vbs−1(k−1) = 8 709563 608667 428235 585802 693265 642469 805517 002156 386113 718666 655956 852142 114070 132769 513810 337524 [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(8110, 262181, F8, 20) (dual of [262181, 262071, 21]-code), using
- construction X with Varšamov bound [i] based on
(91, 91+20, large)-Net in Base 8 — Upper bound on s
There is no (91, 111, large)-net in base 8, because
- 18 times m-reduction [i] would yield (91, 93, large)-net in base 8, but