Best Known (42−11, 42, s)-Nets in Base 8
(42−11, 42, 1639)-Net over F8 — Constructive and digital
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
(42−11, 42, 7699)-Net over F8 — Digital
Digital (31, 42, 7699)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(842, 7699, F8, 11) (dual of [7699, 7657, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(842, 8194, F8, 11) (dual of [8194, 8152, 12]-code), using
- trace code [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]
- trace code [i] based on linear OA(6421, 4097, F64, 11) (dual of [4097, 4076, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(842, 8194, F8, 11) (dual of [8194, 8152, 12]-code), using
(42−11, 42, large)-Net in Base 8 — Upper bound on s
There is no (31, 42, large)-net in base 8, because
- 9 times m-reduction [i] would yield (31, 33, large)-net in base 8, but