Best Known (26, 26+23, s)-Nets in Base 64
(26, 26+23, 373)-Net over F64 — Constructive and digital
Digital (26, 49, 373)-net over F64, using
- 642 times duplication [i] based on digital (24, 47, 373)-net over F64, using
- net defined by OOA [i] based on linear OOA(6447, 373, F64, 23, 23) (dual of [(373, 23), 8532, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- net defined by OOA [i] based on linear OOA(6447, 373, F64, 23, 23) (dual of [(373, 23), 8532, 24]-NRT-code), using
(26, 26+23, 516)-Net in Base 64 — Constructive
(26, 49, 516)-net in base 64, using
- 641 times duplication [i] based on (25, 48, 516)-net in base 64, using
- base change [i] based on digital (13, 36, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 24, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 12, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (13, 36, 516)-net over F256, using
(26, 26+23, 2055)-Net over F64 — Digital
Digital (26, 49, 2055)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6449, 2055, F64, 2, 23) (dual of [(2055, 2), 4061, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6449, 4110, F64, 23) (dual of [4110, 4061, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(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(22) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(6449, 4110, F64, 23) (dual of [4110, 4061, 24]-code), using
(26, 26+23, 5931791)-Net in Base 64 — Upper bound on s
There is no (26, 49, 5931792)-net in base 64, because
- 1 times m-reduction [i] would yield (26, 48, 5931792)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 497 324147 628825 985952 653147 763399 556909 250537 730023 558981 244351 106689 134055 415723 148555 > 6448 [i]