Best Known (147, 162, s)-Nets in Base 8
(147, 162, 3096002)-Net over F8 — Constructive and digital
Digital (147, 162, 3096002)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (41, 48, 699260)-net over F8, using
- net defined by OOA [i] based on linear OOA(848, 699260, F8, 9, 7) (dual of [(699260, 9), 6293292, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(848, 699261, F8, 3, 7) (dual of [(699261, 3), 2097735, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(85, 208, F8, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(85, 208, F8, 2, 3) (dual of [(208, 2), 411, 4]-NRT-code), using
- linear OOA(843, 699053, F8, 3, 7) (dual of [(699053, 3), 2097116, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(843, 2097159, F8, 7) (dual of [2097159, 2097116, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(843, 2097152, F8, 7) (dual of [2097152, 2097109, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding [i] based on linear OA(843, 2097159, F8, 7) (dual of [2097159, 2097116, 8]-code), using
- linear OOA(85, 208, F8, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(848, 699261, F8, 3, 7) (dual of [(699261, 3), 2097735, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(848, 699260, F8, 9, 7) (dual of [(699260, 9), 6293292, 8]-NRT-code), using
- digital (99, 114, 2396742)-net over F8, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- digital (41, 48, 699260)-net over F8, using
(147, 162, 3794849)-Net in Base 8 — Constructive
(147, 162, 3794849)-net in base 8, using
- (u, u+v)-construction [i] based on
- (41, 48, 1398107)-net in base 8, using
- net defined by OOA [i] based on OOA(848, 1398107, S8, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(848, 4194322, S8, 7), using
- trace code [i] based on OA(6424, 2097161, S64, 7), using
- discarding parts of the base [i] based on linear OA(12820, 2097161, F128, 7) (dual of [2097161, 2097141, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1287, 8, F128, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
- dual of repetition code with length 8 [i]
- linear OA(1281, 8, F128, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- Reed–Solomon code RS(127,128) [i]
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- discarding parts of the base [i] based on linear OA(12820, 2097161, F128, 7) (dual of [2097161, 2097141, 8]-code), using
- trace code [i] based on OA(6424, 2097161, S64, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(848, 4194322, S8, 7), using
- net defined by OOA [i] based on OOA(848, 1398107, S8, 7, 7), using
- digital (99, 114, 2396742)-net over F8, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- (41, 48, 1398107)-net in base 8, using
(147, 162, large)-Net over F8 — Digital
Digital (147, 162, large)-net over F8, using
- t-expansion [i] based on digital (144, 162, large)-net over F8, using
- 6 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
- 6 times m-reduction [i] based on digital (144, 168, large)-net over F8, using
(147, 162, large)-Net in Base 8 — Upper bound on s
There is no (147, 162, large)-net in base 8, because
- 13 times m-reduction [i] would yield (147, 149, large)-net in base 8, but