Best Known (80−18, 80, s)-Nets in Base 7
(80−18, 80, 1869)-Net over F7 — Constructive and digital
Digital (62, 80, 1869)-net over F7, using
- 71 times duplication [i] based on digital (61, 79, 1869)-net over F7, using
- net defined by OOA [i] based on linear OOA(779, 1869, F7, 18, 18) (dual of [(1869, 18), 33563, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(779, 16821, F7, 18) (dual of [16821, 16742, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(779, 16825, F7, 18) (dual of [16825, 16746, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(73, 18, F7, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(779, 16825, F7, 18) (dual of [16825, 16746, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(779, 16821, F7, 18) (dual of [16821, 16742, 19]-code), using
- net defined by OOA [i] based on linear OOA(779, 1869, F7, 18, 18) (dual of [(1869, 18), 33563, 19]-NRT-code), using
(80−18, 80, 16827)-Net over F7 — Digital
Digital (62, 80, 16827)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(780, 16827, F7, 18) (dual of [16827, 16747, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(779, 16825, F7, 18) (dual of [16825, 16746, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(73, 18, F7, 2) (dual of [18, 15, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(779, 16826, F7, 17) (dual of [16826, 16747, 18]-code), using Gilbert–Varšamov bound and bm = 779 > Vbs−1(k−1) = 5 521435 871036 956228 846818 155347 502802 355259 224399 985526 718851 466951 [i]
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(779, 16825, F7, 18) (dual of [16825, 16746, 19]-code), using
- construction X with Varšamov bound [i] based on
(80−18, 80, large)-Net in Base 7 — Upper bound on s
There is no (62, 80, large)-net in base 7, because
- 16 times m-reduction [i] would yield (62, 64, large)-net in base 7, but