Best Known (48−19, 48, s)-Nets in Base 64
(48−19, 48, 535)-Net over F64 — Constructive and digital
Digital (29, 48, 535)-net over F64, using
- 641 times duplication [i] based on digital (28, 47, 535)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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 (18, 37, 455)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 455, F64, 19, 19) (dual of [(455, 19), 8608, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using
- net defined by OOA [i] based on linear OOA(6437, 455, F64, 19, 19) (dual of [(455, 19), 8608, 20]-NRT-code), using
- digital (1, 10, 80)-net over F64, using
- (u, u+v)-construction [i] based on
(48−19, 48, 1821)-Net in Base 64 — Constructive
(29, 48, 1821)-net in base 64, using
- 643 times duplication [i] based on (26, 45, 1821)-net in base 64, using
- net defined by OOA [i] based on OOA(6445, 1821, S64, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(6445, 16390, S64, 19), using
- discarding parts of the base [i] based on linear OA(12838, 16390, F128, 19) (dual of [16390, 16352, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- discarding parts of the base [i] based on linear OA(12838, 16390, F128, 19) (dual of [16390, 16352, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on OA(6445, 16390, S64, 19), using
- net defined by OOA [i] based on OOA(6445, 1821, S64, 19, 19), using
(48−19, 48, 7866)-Net over F64 — Digital
Digital (29, 48, 7866)-net over F64, using
(48−19, 48, large)-Net in Base 64 — Upper bound on s
There is no (29, 48, large)-net in base 64, because
- 17 times m-reduction [i] would yield (29, 31, large)-net in base 64, but