Best Known (23, 44, s)-Nets in Base 64
(23, 44, 410)-Net over F64 — Constructive and digital
Digital (23, 44, 410)-net over F64, using
- 642 times duplication [i] based on digital (21, 42, 410)-net over F64, using
- net defined by OOA [i] based on linear OOA(6442, 410, F64, 21, 21) (dual of [(410, 21), 8568, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6442, 4101, F64, 21) (dual of [4101, 4059, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6442, 4102, F64, 21) (dual of [4102, 4060, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6442, 4102, F64, 21) (dual of [4102, 4060, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6442, 4101, F64, 21) (dual of [4101, 4059, 22]-code), using
- net defined by OOA [i] based on linear OOA(6442, 410, F64, 21, 21) (dual of [(410, 21), 8568, 22]-NRT-code), using
(23, 44, 516)-Net in Base 64 — Constructive
(23, 44, 516)-net in base 64, using
- base change [i] based on digital (12, 33, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 22, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 11, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(23, 44, 1956)-Net over F64 — Digital
Digital (23, 44, 1956)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6444, 1956, F64, 2, 21) (dual of [(1956, 2), 3868, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6444, 2054, F64, 2, 21) (dual of [(2054, 2), 4064, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6444, 4108, F64, 21) (dual of [4108, 4064, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- OOA 2-folding [i] based on linear OA(6444, 4108, F64, 21) (dual of [4108, 4064, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(6444, 2054, F64, 2, 21) (dual of [(2054, 2), 4064, 22]-NRT-code), using
(23, 44, 4199611)-Net in Base 64 — Upper bound on s
There is no (23, 44, 4199612)-net in base 64, because
- 1 times m-reduction [i] would yield (23, 43, 4199612)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 463168 440293 089729 123820 025248 505147 452742 402485 992522 179046 604241 642947 795233 > 6443 [i]