Best Known (138, 138+14, s)-Nets in Base 8
(138, 138+14, 3095803)-Net over F8 — Constructive and digital
Digital (138, 152, 3095803)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (39, 46, 699061)-net over F8, using
- net defined by OOA [i] based on linear OOA(846, 699061, F8, 9, 7) (dual of [(699061, 9), 6291503, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(846, 699062, F8, 3, 7) (dual of [(699062, 3), 2097140, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(83, 9, F8, 3, 3) (dual of [(9, 3), 24, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;24,8) [i]
- linear OOA(843, 699053, F8, 3, 7) (dual of [(699053, 3), 2097116, 8]-NRT-code), using
- OOA 3-folding [i] 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
- OOA 3-folding [i] based on linear OA(843, 2097159, F8, 7) (dual of [2097159, 2097116, 8]-code), using
- linear OOA(83, 9, F8, 3, 3) (dual of [(9, 3), 24, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(846, 699062, F8, 3, 7) (dual of [(699062, 3), 2097140, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(846, 699061, F8, 9, 7) (dual of [(699061, 9), 6291503, 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 (39, 46, 699061)-net over F8, using
(138, 138+14, 3794845)-Net in Base 8 — Constructive
(138, 152, 3794845)-net in base 8, using
- (u, u+v)-construction [i] based on
- (39, 46, 1398103)-net in base 8, using
- net defined by OOA [i] based on OOA(846, 1398103, S8, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(846, 4194310, S8, 7), using
- trace code [i] based on OA(6423, 2097155, S64, 7), using
- discarding parts of the base [i] based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- trace code [i] based on OA(6423, 2097155, S64, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(846, 4194310, S8, 7), using
- net defined by OOA [i] based on OOA(846, 1398103, S8, 7, 7), 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
- (39, 46, 1398103)-net in base 8, using
(138, 138+14, large)-Net over F8 — Digital
Digital (138, 152, large)-net over F8, using
- 9 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
(138, 138+14, large)-Net in Base 8 — Upper bound on s
There is no (138, 152, large)-net in base 8, because
- 12 times m-reduction [i] would yield (138, 140, large)-net in base 8, but