Best Known (120, 120+14, s)-Nets in Base 8
(120, 120+14, 2399476)-Net over F8 — Constructive and digital
Digital (120, 134, 2399476)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (21, 28, 2734)-net over F8, using
- net defined by OOA [i] based on linear OOA(828, 2734, F8, 7, 7) (dual of [(2734, 7), 19110, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(828, 8203, F8, 7) (dual of [8203, 8175, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(828, 8204, F8, 7) (dual of [8204, 8176, 8]-code), using
- trace code [i] based on linear OA(6414, 4102, F64, 7) (dual of [4102, 4088, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(649, 4097, F64, 5) (dual of [4097, 4088, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 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
- trace code [i] based on linear OA(6414, 4102, F64, 7) (dual of [4102, 4088, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(828, 8204, F8, 7) (dual of [8204, 8176, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(828, 8203, F8, 7) (dual of [8203, 8175, 8]-code), using
- net defined by OOA [i] based on linear OOA(828, 2734, F8, 7, 7) (dual of [(2734, 7), 19110, 8]-NRT-code), using
- digital (92, 106, 2396742)-net over F8, using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- trace code for nets [i] based on digital (39, 53, 1198371)-net over F64, using
- digital (21, 28, 2734)-net over F8, using
(120, 120+14, large)-Net over F8 — Digital
Digital (120, 134, large)-net over F8, using
- t-expansion [i] based on digital (119, 134, large)-net over F8, using
- 5 times m-reduction [i] based on digital (119, 139, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8139, large, F8, 20) (dual of [large, large−139, 21]-code), using
- 2 times code embedding in larger space [i] based on linear OA(8137, large, F8, 20) (dual of [large, large−137, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 2 times code embedding in larger space [i] based on linear OA(8137, large, F8, 20) (dual of [large, large−137, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8139, large, F8, 20) (dual of [large, large−139, 21]-code), using
- 5 times m-reduction [i] based on digital (119, 139, large)-net over F8, using
(120, 120+14, large)-Net in Base 8 — Upper bound on s
There is no (120, 134, large)-net in base 8, because
- 12 times m-reduction [i] would yield (120, 122, large)-net in base 8, but