Best Known (146−15, 146, s)-Nets in Base 8
(146−15, 146, 2407668)-Net over F8 — Constructive and digital
Digital (131, 146, 2407668)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (25, 32, 10926)-net over F8, using
- net defined by OOA [i] based on linear OOA(832, 10926, F8, 7, 7) (dual of [(10926, 7), 76450, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(832, 32779, F8, 7) (dual of [32779, 32747, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(832, 32780, F8, 7) (dual of [32780, 32748, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(831, 32769, F8, 7) (dual of [32769, 32738, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(821, 32769, F8, 5) (dual of [32769, 32748, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(832, 32780, F8, 7) (dual of [32780, 32748, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(832, 32779, F8, 7) (dual of [32779, 32747, 8]-code), using
- net defined by OOA [i] based on linear OOA(832, 10926, F8, 7, 7) (dual of [(10926, 7), 76450, 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 (25, 32, 10926)-net over F8, using
(146−15, 146, large)-Net over F8 — Digital
Digital (131, 146, large)-net over F8, using
- 7 times m-reduction [i] based on digital (131, 153, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F8, 22) (dual of [large, large−153, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F8, 22) (dual of [large, large−153, 23]-code), using
(146−15, 146, large)-Net in Base 8 — Upper bound on s
There is no (131, 146, large)-net in base 8, because
- 13 times m-reduction [i] would yield (131, 133, large)-net in base 8, but