Best Known (61, 61+14, s)-Nets in Base 128
(61, 61+14, 1897551)-Net over F128 — Constructive and digital
Digital (61, 75, 1897551)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (15, 22, 699180)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (12, 19, 699051)-net over F128, using
- net defined by OOA [i] based on linear OOA(12819, 699051, F128, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12819, 2097154, F128, 7) (dual of [2097154, 2097135, 8]-code), using
- discarding factors / shortening the dual code 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 factors / shortening the dual code based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12819, 2097154, F128, 7) (dual of [2097154, 2097135, 8]-code), using
- net defined by OOA [i] based on linear OOA(12819, 699051, F128, 7, 7) (dual of [(699051, 7), 4893338, 8]-NRT-code), using
- digital (0, 3, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (39, 53, 1198371)-net over F128, using
- net defined by OOA [i] based on linear OOA(12853, 1198371, F128, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(12853, 8388597, F128, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(12853, large, F128, 14) (dual of [large, large−53, 15]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9256395 | 1284−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(12853, large, F128, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(12853, 8388597, F128, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(12853, 1198371, F128, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- digital (15, 22, 699180)-net over F128, using
(61, 61+14, 2397000)-Net in Base 128 — Constructive
(61, 75, 2397000)-net in base 128, using
- (u, u+v)-construction [i] based on
- (15, 22, 2796200)-net in base 128, using
- net defined by OOA [i] based on OOA(12822, 2796200, S128, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(12822, 8388601, S128, 7), using
- discarding factors based on OA(12822, large, S128, 7), using
- discarding parts of the base [i] based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding parts of the base [i] based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- discarding factors based on OA(12822, large, S128, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(12822, 8388601, S128, 7), using
- net defined by OOA [i] based on OOA(12822, 2796200, S128, 7, 7), using
- (39, 53, 1198500)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- (32, 46, 1198371)-net in base 128, using
- net defined by OOA [i] based on OOA(12846, 1198371, S128, 14, 14), using
- OA 7-folding and stacking [i] based on OA(12846, 8388597, S128, 14), using
- discarding factors based on OA(12846, large, S128, 14), using
- discarding parts of the base [i] based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding parts of the base [i] based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- discarding factors based on OA(12846, large, S128, 14), using
- OA 7-folding and stacking [i] based on OA(12846, 8388597, S128, 14), using
- net defined by OOA [i] based on OOA(12846, 1198371, S128, 14, 14), using
- digital (0, 7, 129)-net over F128, using
- (u, u+v)-construction [i] based on
- (15, 22, 2796200)-net in base 128, using
(61, 61+14, large)-Net over F128 — Digital
Digital (61, 75, large)-net over F128, using
- t-expansion [i] based on digital (56, 75, large)-net over F128, using
- 2 times m-reduction [i] based on digital (56, 77, large)-net over F128, using
(61, 61+14, large)-Net in Base 128 — Upper bound on s
There is no (61, 75, large)-net in base 128, because
- 12 times m-reduction [i] would yield (61, 63, large)-net in base 128, but