Best Known (40, 40+13, s)-Nets in Base 8
(40, 40+13, 1367)-Net over F8 — Constructive and digital
Digital (40, 53, 1367)-net over F8, using
- 81 times duplication [i] based on digital (39, 52, 1367)-net over F8, using
- net defined by OOA [i] based on linear OOA(852, 1367, F8, 13, 13) (dual of [(1367, 13), 17719, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(852, 8203, F8, 13) (dual of [8203, 8151, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(852, 8204, F8, 13) (dual of [8204, 8152, 14]-code), using
- trace code [i] based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 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(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([0,6]) ⊂ C([0,5]) [i] based on
- trace code [i] based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(852, 8204, F8, 13) (dual of [8204, 8152, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(852, 8203, F8, 13) (dual of [8203, 8151, 14]-code), using
- net defined by OOA [i] based on linear OOA(852, 1367, F8, 13, 13) (dual of [(1367, 13), 17719, 14]-NRT-code), using
(40, 40+13, 8206)-Net over F8 — Digital
Digital (40, 53, 8206)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(853, 8206, F8, 13) (dual of [8206, 8153, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(852, 8204, F8, 13) (dual of [8204, 8152, 14]-code), using
- trace code [i] based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 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(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([0,6]) ⊂ C([0,5]) [i] based on
- trace code [i] based on linear OA(6426, 4102, F64, 13) (dual of [4102, 4076, 14]-code), using
- linear OA(852, 8205, F8, 12) (dual of [8205, 8153, 13]-code), using Gilbert–Varšamov bound and bm = 852 > Vbs−1(k−1) = 557 667598 560624 431571 643600 935683 193596 222431 [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(852, 8204, F8, 13) (dual of [8204, 8152, 14]-code), using
- construction X with Varšamov bound [i] based on
(40, 40+13, large)-Net in Base 8 — Upper bound on s
There is no (40, 53, large)-net in base 8, because
- 11 times m-reduction [i] would yield (40, 42, large)-net in base 8, but