Best Known (90−18, 90, s)-Nets in Base 64
(90−18, 90, 933156)-Net over F64 — Constructive and digital
Digital (72, 90, 933156)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (12, 21, 1089)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (8, 17, 1024)-net over F64, using
- net defined by OOA [i] based on linear OOA(6417, 1024, F64, 9, 9) (dual of [(1024, 9), 9199, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using
- net defined by OOA [i] based on linear OOA(6417, 1024, F64, 9, 9) (dual of [(1024, 9), 9199, 10]-NRT-code), using
- digital (0, 4, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (51, 69, 932067)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- digital (12, 21, 1089)-net over F64, using
(90−18, 90, 936164)-Net in Base 64 — Constructive
(72, 90, 936164)-net in base 64, using
- (u, u+v)-construction [i] based on
- (12, 21, 4097)-net in base 64, using
- base change [i] based on digital (9, 18, 4097)-net over F128, using
- net defined by OOA [i] based on linear OOA(12818, 4097, F128, 9, 9) (dual of [(4097, 9), 36855, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12818, 16389, F128, 9) (dual of [16389, 16371, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(12818, 16390, F128, 9) (dual of [16390, 16372, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(12813, 16385, F128, 7) (dual of [16385, 16372, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12818, 16390, F128, 9) (dual of [16390, 16372, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(12818, 16389, F128, 9) (dual of [16389, 16371, 10]-code), using
- net defined by OOA [i] based on linear OOA(12818, 4097, F128, 9, 9) (dual of [(4097, 9), 36855, 10]-NRT-code), using
- base change [i] based on digital (9, 18, 4097)-net over F128, using
- digital (51, 69, 932067)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- (12, 21, 4097)-net in base 64, using
(90−18, 90, large)-Net over F64 — Digital
Digital (72, 90, large)-net over F64, using
- 642 times duplication [i] based on digital (70, 88, large)-net over F64, using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
(90−18, 90, large)-Net in Base 64 — Upper bound on s
There is no (72, 90, large)-net in base 64, because
- 16 times m-reduction [i] would yield (72, 74, large)-net in base 64, but