Best Known (32, 32+11, s)-Nets in Base 8
(32, 32+11, 1639)-Net over F8 — Constructive and digital
Digital (32, 43, 1639)-net over F8, using
- 81 times duplication [i] based on digital (31, 42, 1639)-net over F8, using
- net defined by OOA [i] based on linear OOA(842, 1639, F8, 11, 11) (dual of [(1639, 11), 17987, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(842, 8196, F8, 11) (dual of [8196, 8154, 12]-code), using
- trace code [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
- trace code [i] based on linear OA(6421, 4098, F64, 11) (dual of [4098, 4077, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(842, 8196, F8, 11) (dual of [8196, 8154, 12]-code), using
- net defined by OOA [i] based on linear OOA(842, 1639, F8, 11, 11) (dual of [(1639, 11), 17987, 12]-NRT-code), using
(32, 32+11, 8198)-Net over F8 — Digital
Digital (32, 43, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(843, 8198, F8, 11) (dual of [8198, 8155, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(842, 8196, F8, 11) (dual of [8196, 8154, 12]-code), using
- trace code [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
- trace code [i] based on linear OA(6421, 4098, F64, 11) (dual of [4098, 4077, 12]-code), using
- linear OA(842, 8197, F8, 10) (dual of [8197, 8155, 11]-code), using Gilbert–Varšamov bound and bm = 842 > Vbs−1(k−1) = 18 479781 192533 737122 457795 483129 121792 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(842, 8196, F8, 11) (dual of [8196, 8154, 12]-code), using
- construction X with Varšamov bound [i] based on
(32, 32+11, large)-Net in Base 8 — Upper bound on s
There is no (32, 43, large)-net in base 8, because
- 9 times m-reduction [i] would yield (32, 34, large)-net in base 8, but