Best Known (106, 106+13, s)-Nets in Base 8
(106, 106+13, 2797566)-Net over F8 — Constructive and digital
Digital (106, 119, 2797566)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (15, 21, 1366)-net over F8, using
- net defined by OOA [i] based on linear OOA(821, 1366, F8, 6, 6) (dual of [(1366, 6), 8175, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(821, 4098, F8, 6) (dual of [4098, 4077, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(821, 4100, F8, 6) (dual of [4100, 4079, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(821, 4096, F8, 6) (dual of [4096, 4075, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(817, 4096, F8, 5) (dual of [4096, 4079, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(821, 4100, F8, 6) (dual of [4100, 4079, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(821, 4098, F8, 6) (dual of [4098, 4077, 7]-code), using
- net defined by OOA [i] based on linear OOA(821, 1366, F8, 6, 6) (dual of [(1366, 6), 8175, 7]-NRT-code), using
- digital (85, 98, 2796200)-net over F8, using
- net defined by OOA [i] based on linear OOA(898, 2796200, F8, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(898, 8388601, F8, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(898, 8388602, F8, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- trace code [i] based on linear OOA(6449, 4194301, F64, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6449, 8388602, F64, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(6449, 8388602, F64, 13) (dual of [8388602, 8388553, 14]-code), using
- trace code [i] based on linear OOA(6449, 4194301, F64, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(898, 8388602, F8, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(898, 8388601, F8, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(898, 2796200, F8, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- digital (15, 21, 1366)-net over F8, using
(106, 106+13, large)-Net over F8 — Digital
Digital (106, 119, large)-net over F8, using
- 5 times m-reduction [i] based on digital (106, 124, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8124, large, F8, 18) (dual of [large, large−124, 19]-code), using
- 3 times code embedding in larger space [i] based on linear OA(8121, large, F8, 18) (dual of [large, large−121, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 3 times code embedding in larger space [i] based on linear OA(8121, large, F8, 18) (dual of [large, large−121, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8124, large, F8, 18) (dual of [large, large−124, 19]-code), using
(106, 106+13, large)-Net in Base 8 — Upper bound on s
There is no (106, 119, large)-net in base 8, because
- 11 times m-reduction [i] would yield (106, 108, large)-net in base 8, but