Best Known (141, 141+17, s)-Nets in Base 8
(141, 141+17, 2098173)-Net over F8 — Constructive and digital
Digital (141, 158, 2098173)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (20, 28, 1023)-net over F8, using
- net defined by OOA [i] based on linear OOA(828, 1023, F8, 8, 8) (dual of [(1023, 8), 8156, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(828, 4092, F8, 8) (dual of [4092, 4064, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(828, 4095, F8, 8) (dual of [4095, 4067, 9]-code), using
- 1 times truncation [i] based on linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 1 times truncation [i] based on linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(828, 4095, F8, 8) (dual of [4095, 4067, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(828, 4092, F8, 8) (dual of [4092, 4064, 9]-code), using
- net defined by OOA [i] based on linear OOA(828, 1023, F8, 8, 8) (dual of [(1023, 8), 8156, 9]-NRT-code), using
- digital (113, 130, 2097150)-net over F8, using
- net defined by OOA [i] based on linear OOA(8130, 2097150, F8, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8130, 8388601, F8, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8130, 8388602, F8, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
- trace code [i] based on linear OOA(6465, 4194301, F64, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6465, 8388602, F64, 17) (dual of [8388602, 8388537, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- OOA 2-folding [i] based on linear OA(6465, 8388602, F64, 17) (dual of [8388602, 8388537, 18]-code), using
- trace code [i] based on linear OOA(6465, 4194301, F64, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8130, 8388602, F8, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8130, 8388601, F8, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8130, 2097150, F8, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
- digital (20, 28, 1023)-net over F8, using
(141, 141+17, large)-Net over F8 — Digital
Digital (141, 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
(141, 141+17, large)-Net in Base 8 — Upper bound on s
There is no (141, 158, large)-net in base 8, because
- 15 times m-reduction [i] would yield (141, 143, large)-net in base 8, but