Best Known (111, 111+26, s)-Nets in Base 8
(111, 111+26, 20166)-Net over F8 — Constructive and digital
Digital (111, 137, 20166)-net over F8, using
- 81 times duplication [i] based on digital (110, 136, 20166)-net over F8, using
- net defined by OOA [i] based on linear OOA(8136, 20166, F8, 26, 26) (dual of [(20166, 26), 524180, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8136, 262158, F8, 26) (dual of [262158, 262022, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8136, 262159, F8, 26) (dual of [262159, 262023, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(8133, 262144, F8, 26) (dual of [262144, 262011, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8121, 262144, F8, 23) (dual of [262144, 262023, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(83, 15, F8, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8136, 262159, F8, 26) (dual of [262159, 262023, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8136, 262158, F8, 26) (dual of [262158, 262022, 27]-code), using
- net defined by OOA [i] based on linear OOA(8136, 20166, F8, 26, 26) (dual of [(20166, 26), 524180, 27]-NRT-code), using
(111, 111+26, 183544)-Net over F8 — Digital
Digital (111, 137, 183544)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8137, 183544, F8, 26) (dual of [183544, 183407, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, 262166, F8, 26) (dual of [262166, 262029, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(8133, 262144, F8, 26) (dual of [262144, 262011, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8115, 262144, F8, 22) (dual of [262144, 262029, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8137, 262166, F8, 26) (dual of [262166, 262029, 27]-code), using
(111, 111+26, large)-Net in Base 8 — Upper bound on s
There is no (111, 137, large)-net in base 8, because
- 24 times m-reduction [i] would yield (111, 113, large)-net in base 8, but