Best Known (18, 37, s)-Nets in Base 64
(18, 37, 455)-Net over F64 — Constructive and digital
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
(18, 37, 1259)-Net over F64 — Digital
Digital (18, 37, 1259)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6437, 1259, F64, 3, 19) (dual of [(1259, 3), 3740, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6437, 1366, F64, 3, 19) (dual of [(1366, 3), 4061, 20]-NRT-code), using
- OOA 3-folding [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
- OOA 3-folding [i] based on linear OA(6437, 4098, F64, 19) (dual of [4098, 4061, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(6437, 1366, F64, 3, 19) (dual of [(1366, 3), 4061, 20]-NRT-code), using
(18, 37, 1104407)-Net in Base 64 — Upper bound on s
There is no (18, 37, 1104408)-net in base 64, because
- 1 times m-reduction [i] would yield (18, 36, 1104408)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 105313 117540 740613 715489 659059 362596 511919 372910 210320 115967 035162 > 6436 [i]