Best Known (158−15, 158, s)-Nets in Base 8
(158−15, 158, 3095798)-Net over F8 — Constructive and digital
Digital (143, 158, 3095798)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (37, 44, 699056)-net over F8, using
- net defined by OOA [i] based on linear OOA(844, 699056, F8, 7, 7) (dual of [(699056, 7), 4893348, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(844, 2097169, F8, 7) (dual of [2097169, 2097125, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(843, 2097153, F8, 7) (dual of [2097153, 2097110, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(829, 2097153, F8, 5) (dual of [2097153, 2097124, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(815, 16, F8, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,8)), using
- dual of repetition code with length 16 [i]
- linear OA(81, 16, F8, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(844, 2097169, F8, 7) (dual of [2097169, 2097125, 8]-code), using
- net defined by OOA [i] based on linear OOA(844, 699056, F8, 7, 7) (dual of [(699056, 7), 4893348, 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 (37, 44, 699056)-net over F8, using
(158−15, 158, large)-Net over F8 — Digital
Digital (143, 158, large)-net over F8, using
- t-expansion [i] based on digital (138, 158, large)-net over F8, using
- 3 times m-reduction [i] based on digital (138, 161, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8161, large, F8, 23) (dual of [large, large−161, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8161, large, F8, 23) (dual of [large, large−161, 24]-code), using
- 3 times m-reduction [i] based on digital (138, 161, large)-net over F8, using
(158−15, 158, large)-Net in Base 8 — Upper bound on s
There is no (143, 158, large)-net in base 8, because
- 13 times m-reduction [i] would yield (143, 145, large)-net in base 8, but