Best Known (43−18, 43, s)-Nets in Base 64
(43−18, 43, 458)-Net over F64 — Constructive and digital
Digital (25, 43, 458)-net over F64, using
- net defined by OOA [i] based on linear OOA(6443, 458, F64, 18, 18) (dual of [(458, 18), 8201, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6443, 4122, F64, 18) (dual of [4122, 4079, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(8) [i] based on
- 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(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(648, 26, F64, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,64)), using
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- Reed–Solomon code RS(56,64) [i]
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- construction X applied to Ce(17) ⊂ Ce(8) [i] based on
- OA 9-folding and stacking [i] based on linear OA(6443, 4122, F64, 18) (dual of [4122, 4079, 19]-code), using
(43−18, 43, 1821)-Net in Base 64 — Constructive
(25, 43, 1821)-net in base 64, using
- 641 times duplication [i] based on (24, 42, 1821)-net in base 64, using
- base change [i] based on digital (18, 36, 1821)-net over F128, using
- net defined by OOA [i] based on linear OOA(12836, 1821, F128, 18, 18) (dual of [(1821, 18), 32742, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(12835, 16384, F128, 18) (dual of [16384, 16349, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(12831, 16384, F128, 16) (dual of [16384, 16353, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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 Ce(17) ⊂ Ce(15) [i] based on
- OA 9-folding and stacking [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
- net defined by OOA [i] based on linear OOA(12836, 1821, F128, 18, 18) (dual of [(1821, 18), 32742, 19]-NRT-code), using
- base change [i] based on digital (18, 36, 1821)-net over F128, using
(43−18, 43, 4696)-Net over F64 — Digital
Digital (25, 43, 4696)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6443, 4696, F64, 18) (dual of [4696, 4653, 19]-code), using
- 590 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 15 times 0, 1, 50 times 0, 1, 143 times 0, 1, 373 times 0) [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(6433, 4096, F64, 17) (dual of [4096, 4063, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- 590 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 15 times 0, 1, 50 times 0, 1, 143 times 0, 1, 373 times 0) [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
(43−18, 43, large)-Net in Base 64 — Upper bound on s
There is no (25, 43, large)-net in base 64, because
- 16 times m-reduction [i] would yield (25, 27, large)-net in base 64, but