Best Known (40−15, 40, s)-Nets in Base 64
(40−15, 40, 730)-Net over F64 — Constructive and digital
Digital (25, 40, 730)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 145)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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 (1, 8, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (0, 3, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (14, 29, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using
- net defined by OOA [i] based on linear OOA(6429, 585, F64, 15, 15) (dual of [(585, 15), 8746, 16]-NRT-code), using
- digital (4, 11, 145)-net over F64, using
(40−15, 40, 9363)-Net in Base 64 — Constructive
(25, 40, 9363)-net in base 64, using
- base change [i] based on digital (15, 30, 9363)-net over F256, using
- net defined by OOA [i] based on linear OOA(25630, 9363, F256, 15, 15) (dual of [(9363, 15), 140415, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25630, 65542, F256, 15) (dual of [65542, 65512, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(25630, 65542, F256, 15) (dual of [65542, 65512, 16]-code), using
- net defined by OOA [i] based on linear OOA(25630, 9363, F256, 15, 15) (dual of [(9363, 15), 140415, 16]-NRT-code), using
(40−15, 40, 13895)-Net over F64 — Digital
Digital (25, 40, 13895)-net over F64, using
(40−15, 40, 16385)-Net in Base 64
(25, 40, 16385)-net in base 64, using
- base change [i] based on digital (15, 30, 16385)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25630, 16385, F256, 4, 15) (dual of [(16385, 4), 65510, 16]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25630, 65540, F256, 15) (dual of [65540, 65510, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25630, 65542, F256, 15) (dual of [65542, 65512, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25630, 65542, F256, 15) (dual of [65542, 65512, 16]-code), using
- OOA 4-folding [i] based on linear OA(25630, 65540, F256, 15) (dual of [65540, 65510, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25630, 16385, F256, 4, 15) (dual of [(16385, 4), 65510, 16]-NRT-code), using
(40−15, 40, large)-Net in Base 64 — Upper bound on s
There is no (25, 40, large)-net in base 64, because
- 13 times m-reduction [i] would yield (25, 27, large)-net in base 64, but