Best Known (22, 41, s)-Nets in Base 64
(22, 41, 456)-Net over F64 — Constructive and digital
Digital (22, 41, 456)-net over F64, using
- 641 times duplication [i] based on digital (21, 40, 456)-net over F64, using
- net defined by OOA [i] based on linear OOA(6440, 456, F64, 19, 19) (dual of [(456, 19), 8624, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6440, 4105, F64, 19) (dual of [4105, 4065, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 4108, F64, 19) (dual of [4108, 4068, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [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(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 4108, F64, 19) (dual of [4108, 4068, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6440, 4105, F64, 19) (dual of [4105, 4065, 20]-code), using
- net defined by OOA [i] based on linear OOA(6440, 456, F64, 19, 19) (dual of [(456, 19), 8624, 20]-NRT-code), using
(22, 41, 516)-Net in Base 64 — Constructive
(22, 41, 516)-net in base 64, using
- 641 times duplication [i] based on (21, 40, 516)-net in base 64, using
- base change [i] based on digital (11, 30, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 20, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 10, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (11, 30, 516)-net over F256, using
(22, 41, 2055)-Net over F64 — Digital
Digital (22, 41, 2055)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6441, 2055, F64, 2, 19) (dual of [(2055, 2), 4069, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6441, 4110, F64, 19) (dual of [4110, 4069, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [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(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(6441, 4110, F64, 19) (dual of [4110, 4069, 20]-code), using
(22, 41, 7012569)-Net in Base 64 — Upper bound on s
There is no (22, 41, 7012570)-net in base 64, because
- 1 times m-reduction [i] would yield (22, 40, 7012570)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 1 766847 662816 624204 091598 473225 754788 233202 750633 286166 692592 647128 383833 > 6440 [i]