Best Known (34, 34+11, s)-Nets in Base 8
(34, 34+11, 1640)-Net over F8 — Constructive and digital
Digital (34, 45, 1640)-net over F8, using
- 81 times duplication [i] based on digital (33, 44, 1640)-net over F8, using
- net defined by OOA [i] based on linear OOA(844, 1640, F8, 11, 11) (dual of [(1640, 11), 17996, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(844, 8201, F8, 11) (dual of [8201, 8157, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 8204, F8, 11) (dual of [8204, 8160, 12]-code), using
- trace code [i] based on linear OA(6422, 4102, F64, 11) (dual of [4102, 4080, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [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(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- trace code [i] based on linear OA(6422, 4102, F64, 11) (dual of [4102, 4080, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 8204, F8, 11) (dual of [8204, 8160, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(844, 8201, F8, 11) (dual of [8201, 8157, 12]-code), using
- net defined by OOA [i] based on linear OOA(844, 1640, F8, 11, 11) (dual of [(1640, 11), 17996, 12]-NRT-code), using
(34, 34+11, 8206)-Net over F8 — Digital
Digital (34, 45, 8206)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(845, 8206, F8, 11) (dual of [8206, 8161, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(844, 8204, F8, 11) (dual of [8204, 8160, 12]-code), using
- trace code [i] based on linear OA(6422, 4102, F64, 11) (dual of [4102, 4080, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [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(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- trace code [i] based on linear OA(6422, 4102, F64, 11) (dual of [4102, 4080, 12]-code), using
- linear OA(844, 8205, F8, 10) (dual of [8205, 8161, 11]-code), using Gilbert–Varšamov bound and bm = 844 > Vbs−1(k−1) = 18 642834 203705 403879 025272 696996 690601 [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(844, 8204, F8, 11) (dual of [8204, 8160, 12]-code), using
- construction X with Varšamov bound [i] based on
(34, 34+11, large)-Net in Base 8 — Upper bound on s
There is no (34, 45, large)-net in base 8, because
- 9 times m-reduction [i] would yield (34, 36, large)-net in base 8, but