Best Known (23, 23+17, s)-Nets in Base 64
(23, 23+17, 514)-Net over F64 — Constructive and digital
Digital (23, 40, 514)-net over F64, using
- 642 times duplication [i] based on digital (21, 38, 514)-net over F64, using
- net defined by OOA [i] based on linear OOA(6438, 514, F64, 17, 17) (dual of [(514, 17), 8700, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6438, 4113, F64, 17) (dual of [4113, 4075, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6438, 4114, F64, 17) (dual of [4114, 4076, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- 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(6421, 4097, F64, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6438, 4114, F64, 17) (dual of [4114, 4076, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6438, 4113, F64, 17) (dual of [4113, 4075, 18]-code), using
- net defined by OOA [i] based on linear OOA(6438, 514, F64, 17, 17) (dual of [(514, 17), 8700, 18]-NRT-code), using
(23, 23+17, 2048)-Net in Base 64 — Constructive
(23, 40, 2048)-net in base 64, using
- 641 times duplication [i] based on (22, 39, 2048)-net in base 64, using
- net defined by OOA [i] based on OOA(6439, 2048, S64, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6439, 16385, S64, 17), using
- discarding factors based on OA(6439, 16386, S64, 17), using
- discarding parts of the base [i] based on linear OA(12833, 16386, F128, 17) (dual of [16386, 16353, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(12833, 16384, F128, 17) (dual of [16384, 16351, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(12833, 16386, F128, 17) (dual of [16386, 16353, 18]-code), using
- discarding factors based on OA(6439, 16386, S64, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6439, 16385, S64, 17), using
- net defined by OOA [i] based on OOA(6439, 2048, S64, 17, 17), using
(23, 23+17, 4408)-Net over F64 — Digital
Digital (23, 40, 4408)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6440, 4408, F64, 17) (dual of [4408, 4368, 18]-code), using
- 303 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 18 times 0, 1, 69 times 0, 1, 208 times 0) [i] based on linear OA(6433, 4098, F64, 17) (dual of [4098, 4065, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- 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(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- 303 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 18 times 0, 1, 69 times 0, 1, 208 times 0) [i] based on linear OA(6433, 4098, F64, 17) (dual of [4098, 4065, 18]-code), using
(23, 23+17, large)-Net in Base 64 — Upper bound on s
There is no (23, 40, large)-net in base 64, because
- 15 times m-reduction [i] would yield (23, 25, large)-net in base 64, but