Best Known (157−15, 157, s)-Nets in Base 8
(157−15, 157, 3095794)-Net over F8 — Constructive and digital
Digital (142, 157, 3095794)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (36, 43, 699052)-net over F8, using
- net defined by OOA [i] based on linear OOA(843, 699052, F8, 7, 7) (dual of [(699052, 7), 4893321, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(843, 2097157, F8, 7) (dual of [2097157, 2097114, 8]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(843, 2097159, F8, 7) (dual of [2097159, 2097116, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(843, 2097157, F8, 7) (dual of [2097157, 2097114, 8]-code), using
- net defined by OOA [i] based on linear OOA(843, 699052, F8, 7, 7) (dual of [(699052, 7), 4893321, 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 (36, 43, 699052)-net over F8, using
(157−15, 157, large)-Net over F8 — Digital
Digital (142, 157, large)-net over F8, using
- t-expansion [i] based on digital (138, 157, large)-net over F8, using
- 4 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
- 4 times m-reduction [i] based on digital (138, 161, large)-net over F8, using
(157−15, 157, large)-Net in Base 8 — Upper bound on s
There is no (142, 157, large)-net in base 8, because
- 13 times m-reduction [i] would yield (142, 144, large)-net in base 8, but