Best Known (109, 109+20, s)-Nets in Base 8
(109, 109+20, 209719)-Net over F8 — Constructive and digital
Digital (109, 129, 209719)-net over F8, using
- 82 times duplication [i] based on digital (107, 127, 209719)-net over F8, using
- net defined by OOA [i] based on linear OOA(8127, 209719, F8, 20, 20) (dual of [(209719, 20), 4194253, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8127, 2097190, F8, 20) (dual of [2097190, 2097063, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8127, 2097194, F8, 20) (dual of [2097194, 2097067, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(8120, 2097152, F8, 20) (dual of [2097152, 2097032, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(885, 2097152, F8, 14) (dual of [2097152, 2097067, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 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(8127, 2097194, F8, 20) (dual of [2097194, 2097067, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8127, 2097190, F8, 20) (dual of [2097190, 2097063, 21]-code), using
- net defined by OOA [i] based on linear OOA(8127, 209719, F8, 20, 20) (dual of [(209719, 20), 4194253, 21]-NRT-code), using
(109, 109+20, 2097198)-Net over F8 — Digital
Digital (109, 129, 2097198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8129, 2097198, F8, 20) (dual of [2097198, 2097069, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8127, 2097194, F8, 20) (dual of [2097194, 2097067, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(8120, 2097152, F8, 20) (dual of [2097152, 2097032, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(885, 2097152, F8, 14) (dual of [2097152, 2097067, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 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(8127, 2097196, F8, 19) (dual of [2097196, 2097069, 20]-code), using Gilbert–Varšamov bound and bm = 8127 > Vbs−1(k−1) = 156635 859279 981246 217339 373067 149265 044089 906141 032582 079922 596287 270996 627614 758824 563159 663990 799661 872676 746730 [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(8127, 2097194, F8, 20) (dual of [2097194, 2097067, 21]-code), using
- construction X with Varšamov bound [i] based on
(109, 109+20, large)-Net in Base 8 — Upper bound on s
There is no (109, 129, large)-net in base 8, because
- 18 times m-reduction [i] would yield (109, 111, large)-net in base 8, but