Best Known (15, 26, s)-Nets in Base 64
(15, 26, 884)-Net over F64 — Constructive and digital
Digital (15, 26, 884)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (10, 21, 819)-net over F64, using
- net defined by OOA [i] based on linear OOA(6421, 819, F64, 11, 11) (dual of [(819, 11), 8988, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [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]
- OOA 5-folding and stacking with additional row [i] based on linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using
- net defined by OOA [i] based on linear OOA(6421, 819, F64, 11, 11) (dual of [(819, 11), 8988, 12]-NRT-code), using
- digital (0, 5, 65)-net over F64, using
(15, 26, 3277)-Net in Base 64 — Constructive
(15, 26, 3277)-net in base 64, using
- 641 times duplication [i] based on (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
- net defined by OOA [i] based on OOA(6425, 3277, S64, 11, 11), using
(15, 26, 4529)-Net over F64 — Digital
Digital (15, 26, 4529)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6426, 4529, F64, 11) (dual of [4529, 4503, 12]-code), using
- 426 step Varšamov–Edel lengthening with (ri) = (3, 8 times 0, 1, 68 times 0, 1, 347 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
- 426 step Varšamov–Edel lengthening with (ri) = (3, 8 times 0, 1, 68 times 0, 1, 347 times 0) [i] based on linear OA(6421, 4098, F64, 11) (dual of [4098, 4077, 12]-code), using
(15, 26, large)-Net in Base 64 — Upper bound on s
There is no (15, 26, large)-net in base 64, because
- 9 times m-reduction [i] would yield (15, 17, large)-net in base 64, but