Best Known (42−18, 42, s)-Nets in Base 64
(42−18, 42, 457)-Net over F64 — Constructive and digital
Digital (24, 42, 457)-net over F64, using
- t-expansion [i] based on digital (23, 42, 457)-net over F64, using
- net defined by OOA [i] based on linear OOA(6442, 457, F64, 19, 19) (dual of [(457, 19), 8641, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6442, 4114, F64, 19) (dual of [4114, 4072, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [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(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(6442, 4114, F64, 19) (dual of [4114, 4072, 20]-code), using
- net defined by OOA [i] based on linear OOA(6442, 457, F64, 19, 19) (dual of [(457, 19), 8641, 20]-NRT-code), using
(42−18, 42, 1821)-Net in Base 64 — Constructive
(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
(42−18, 42, 4321)-Net over F64 — Digital
Digital (24, 42, 4321)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6442, 4321, F64, 18) (dual of [4321, 4279, 19]-code), using
- 216 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 15 times 0, 1, 50 times 0, 1, 143 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
- 216 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 15 times 0, 1, 50 times 0, 1, 143 times 0) [i] based on linear OA(6435, 4098, F64, 18) (dual of [4098, 4063, 19]-code), using
(42−18, 42, 4406)-Net in Base 64
(24, 42, 4406)-net in base 64, using
- base change [i] based on digital (18, 36, 4406)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12836, 4406, F128, 3, 18) (dual of [(4406, 3), 13182, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12836, 5463, F128, 3, 18) (dual of [(5463, 3), 16353, 19]-NRT-code), using
- OOA 3-folding [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
- OOA 3-folding [i] based on linear OA(12836, 16389, F128, 18) (dual of [16389, 16353, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(12836, 5463, F128, 3, 18) (dual of [(5463, 3), 16353, 19]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12836, 4406, F128, 3, 18) (dual of [(4406, 3), 13182, 19]-NRT-code), using
(42−18, 42, large)-Net in Base 64 — Upper bound on s
There is no (24, 42, large)-net in base 64, because
- 16 times m-reduction [i] would yield (24, 26, large)-net in base 64, but