Best Known (50−11, 50, s)-Nets in Base 7
(50−11, 50, 3365)-Net over F7 — Constructive and digital
Digital (39, 50, 3365)-net over F7, using
- net defined by OOA [i] based on linear OOA(750, 3365, F7, 11, 11) (dual of [(3365, 11), 36965, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(750, 16826, F7, 11) (dual of [16826, 16776, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(10) ⊂ Ce(7) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(750, 16826, F7, 11) (dual of [16826, 16776, 12]-code), using
(50−11, 50, 16827)-Net over F7 — Digital
Digital (39, 50, 16827)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(750, 16827, F7, 11) (dual of [16827, 16777, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(10) ⊂ Ce(7) [i] based on
- linear OA(749, 16826, F7, 10) (dual of [16826, 16777, 11]-code), using Gilbert–Varšamov bound and bm = 749 > Vbs−1(k−1) = 2994 354468 231915 442866 515084 866976 097991 [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(749, 16825, F7, 11) (dual of [16825, 16776, 12]-code), using
- construction X with Varšamov bound [i] based on
(50−11, 50, large)-Net in Base 7 — Upper bound on s
There is no (39, 50, large)-net in base 7, because
- 9 times m-reduction [i] would yield (39, 41, large)-net in base 7, but