Best Known (45−19, 45, s)-Nets in Base 64
(45−19, 45, 457)-Net over F64 — Constructive and digital
Digital (26, 45, 457)-net over F64, using
- 643 times duplication [i] based on digital (23, 42, 457)-net over F64, using
- net defined by OOA [i] based on linear OOA(6442, 457, F64, 19, 19) (dual of [(457, 19), 8641, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6442, 4114, F64, 19) (dual of [4114, 4072, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(6442, 4114, F64, 19) (dual of [4114, 4072, 20]-code), using
- net defined by OOA [i] based on linear OOA(6442, 457, F64, 19, 19) (dual of [(457, 19), 8641, 20]-NRT-code), using
(45−19, 45, 1821)-Net in Base 64 — Constructive
(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
(45−19, 45, 4523)-Net over F64 — Digital
Digital (26, 45, 4523)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6445, 4523, F64, 19) (dual of [4523, 4478, 20]-code), using
- 417 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 0, 1, 11 times 0, 1, 34 times 0, 1, 99 times 0, 1, 265 times 0) [i] based on linear OA(6437, 4098, F64, 19) (dual of [4098, 4061, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [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]
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- 417 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 0, 1, 11 times 0, 1, 34 times 0, 1, 99 times 0, 1, 265 times 0) [i] based on linear OA(6437, 4098, F64, 19) (dual of [4098, 4061, 20]-code), using
(45−19, 45, large)-Net in Base 64 — Upper bound on s
There is no (26, 45, large)-net in base 64, because
- 17 times m-reduction [i] would yield (26, 28, large)-net in base 64, but