Best Known (58, 72, s)-Nets in Base 128
(58, 72, 1897422)-Net over F128 — Constructive and digital
Digital (58, 72, 1897422)-net over F128, using
- (u, u+v)-construction [i] based on
- 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 (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 (12, 19, 699051)-net over F128, using
(58, 72, 2396999)-Net in Base 128 — Constructive
(58, 72, 2396999)-net in base 128, using
- base change [i] based on digital (49, 63, 2396999)-net over F256, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- digital (12, 19, 1198371)-net over F256, using
- s-reduction based on digital (12, 19, 2796200)-net over F256, using
- net defined by OOA [i] based on linear OOA(25619, 2796200, F256, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25619, 2796200, F256, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- discarding factors / shortening the dual code 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 factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- appending kth column [i] based on linear OOA(25619, 2796200, F256, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25619, 2796200, F256, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
- s-reduction based on digital (12, 19, 2796200)-net over F256, using
- digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code 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 factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- digital (0, 4, 257)-net over F256, using
- generalized (u, u+v)-construction [i] based on
(58, 72, large)-Net over F128 — Digital
Digital (58, 72, large)-net over F128, using
- t-expansion [i] based on digital (56, 72, large)-net over F128, using
- 5 times m-reduction [i] based on digital (56, 77, large)-net over F128, using
(58, 72, large)-Net in Base 128 — Upper bound on s
There is no (58, 72, large)-net in base 128, because
- 12 times m-reduction [i] would yield (58, 60, large)-net in base 128, but