Best Known (101, 101+11, s)-Nets in Base 8
(101, 101+11, 4404024)-Net over F8 — Constructive and digital
Digital (101, 112, 4404024)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (25, 30, 1048584)-net over F8, using
- net defined by OOA [i] based on linear OOA(830, 1048584, F8, 5, 5) (dual of [(1048584, 5), 5242890, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(830, 2097169, F8, 5) (dual of [2097169, 2097139, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- 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, 2097153, F8, 3) (dual of [2097153, 2097138, 4]-code or 2097153-cap in PG(14,8)), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (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([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(830, 2097169, F8, 5) (dual of [2097169, 2097139, 6]-code), using
- net defined by OOA [i] based on linear OOA(830, 1048584, F8, 5, 5) (dual of [(1048584, 5), 5242890, 6]-NRT-code), using
- digital (71, 82, 3355440)-net over F8, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F64, using
- digital (25, 30, 1048584)-net over F8, using
(101, 101+11, large)-Net over F8 — Digital
Digital (101, 112, large)-net over F8, using
- t-expansion [i] based on digital (100, 112, large)-net over F8, using
- 5 times m-reduction [i] based on digital (100, 117, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8117, large, F8, 17) (dual of [large, large−117, 18]-code), using
- 4 times code embedding in larger space [i] based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 4 times code embedding in larger space [i] based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8117, large, F8, 17) (dual of [large, large−117, 18]-code), using
- 5 times m-reduction [i] based on digital (100, 117, large)-net over F8, using
(101, 101+11, large)-Net in Base 8 — Upper bound on s
There is no (101, 112, large)-net in base 8, because
- 9 times m-reduction [i] would yield (101, 103, large)-net in base 8, but