Best Known (163−13, 163, s)-Nets in Base 8
(163−13, 163, 6640988)-Net over F8 — Constructive and digital
Digital (150, 163, 6640988)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (20, 24, 1048588)-net over F8, using
- net defined by OOA [i] based on linear OOA(824, 1048588, F8, 4, 4) (dual of [(1048588, 4), 4194328, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(824, 1048588, F8, 3, 4) (dual of [(1048588, 3), 3145740, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(82, 9, F8, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;25,8) [i]
- linear OOA(822, 1048579, F8, 3, 4) (dual of [(1048579, 3), 3145715, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(822, 2097158, F8, 4) (dual of [2097158, 2097136, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(822, 2097159, F8, 4) (dual of [2097159, 2097137, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(822, 2097152, F8, 4) (dual of [2097152, 2097130, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(815, 2097152, F8, 3) (dual of [2097152, 2097137, 4]-code or 2097152-cap in PG(14,8)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(822, 2097159, F8, 4) (dual of [2097159, 2097137, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(822, 2097158, F8, 4) (dual of [2097158, 2097136, 5]-code), using
- linear OOA(82, 9, F8, 3, 2) (dual of [(9, 3), 25, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(824, 1048588, F8, 3, 4) (dual of [(1048588, 3), 3145740, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(824, 1048588, F8, 4, 4) (dual of [(1048588, 4), 4194328, 5]-NRT-code), using
- digital (35, 41, 2796200)-net over F8, using
- s-reduction based on digital (35, 41, 2796201)-net over F8, using
- net defined by OOA [i] based on linear OOA(841, 2796201, F8, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(841, large, F8, 6) (dual of [large, large−41, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(841, large, F8, 6) (dual of [large, large−41, 7]-code), using
- net defined by OOA [i] based on linear OOA(841, 2796201, F8, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- s-reduction based on digital (35, 41, 2796201)-net over F8, 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 (20, 24, 1048588)-net over F8, using
(163−13, 163, large)-Net over F8 — Digital
Digital (150, 163, large)-net over F8, using
- t-expansion [i] based on digital (144, 163, large)-net over F8, using
- 5 times m-reduction [i] based on digital (144, 168, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8168, large, F8, 24) (dual of [large, large−168, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8168, large, F8, 24) (dual of [large, large−168, 25]-code), using
- 5 times m-reduction [i] based on digital (144, 168, large)-net over F8, using
(163−13, 163, large)-Net in Base 8 — Upper bound on s
There is no (150, 163, large)-net in base 8, because
- 11 times m-reduction [i] would yield (150, 152, large)-net in base 8, but