Best Known (25−11, 25, s)-Nets in Base 64
(25−11, 25, 821)-Net over F64 — Constructive and digital
Digital (14, 25, 821)-net over F64, using
- 641 times duplication [i] based on digital (13, 24, 821)-net over F64, using
- net defined by OOA [i] based on linear OOA(6424, 821, F64, 11, 11) (dual of [(821, 11), 9007, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6424, 4106, F64, 11) (dual of [4106, 4082, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6424, 4108, F64, 11) (dual of [4108, 4084, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- 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(6413, 4097, F64, 7) (dual of [4097, 4084, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,5]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6424, 4108, F64, 11) (dual of [4108, 4084, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6424, 4106, F64, 11) (dual of [4106, 4082, 12]-code), using
- net defined by OOA [i] based on linear OOA(6424, 821, F64, 11, 11) (dual of [(821, 11), 9007, 12]-NRT-code), using
(25−11, 25, 3277)-Net in Base 64 — Constructive
(14, 25, 3277)-net in base 64, using
- net defined by OOA [i] based on OOA(6425, 3277, S64, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6425, 16386, S64, 11), using
- discarding parts of the base [i] based on linear OA(12821, 16386, F128, 11) (dual of [16386, 16365, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(12821, 16384, F128, 11) (dual of [16384, 16363, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- discarding parts of the base [i] based on linear OA(12821, 16386, F128, 11) (dual of [16386, 16365, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on OA(6425, 16386, S64, 11), using
(25−11, 25, 4180)-Net over F64 — Digital
Digital (14, 25, 4180)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6425, 4180, F64, 11) (dual of [4180, 4155, 12]-code), using
- 78 step Varšamov–Edel lengthening with (ri) = (3, 8 times 0, 1, 68 times 0) [i] based on linear OA(6421, 4098, F64, 11) (dual of [4098, 4077, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- 78 step Varšamov–Edel lengthening with (ri) = (3, 8 times 0, 1, 68 times 0) [i] based on linear OA(6421, 4098, F64, 11) (dual of [4098, 4077, 12]-code), using
(25−11, 25, large)-Net in Base 64 — Upper bound on s
There is no (14, 25, large)-net in base 64, because
- 9 times m-reduction [i] would yield (14, 16, large)-net in base 64, but