Best Known (65−18, 65, s)-Nets in Base 7
(65−18, 65, 268)-Net over F7 — Constructive and digital
Digital (47, 65, 268)-net over F7, using
- 72 times duplication [i] based on digital (45, 63, 268)-net over F7, using
- net defined by OOA [i] based on linear OOA(763, 268, F7, 18, 18) (dual of [(268, 18), 4761, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(763, 2412, F7, 18) (dual of [2412, 2349, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- linear OA(761, 2401, F7, 18) (dual of [2401, 2340, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(753, 2401, F7, 16) (dual of [2401, 2348, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(71, 10, F7, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- OA 9-folding and stacking [i] based on linear OA(763, 2412, F7, 18) (dual of [2412, 2349, 19]-code), using
- net defined by OOA [i] based on linear OOA(763, 268, F7, 18, 18) (dual of [(268, 18), 4761, 19]-NRT-code), using
(65−18, 65, 2452)-Net over F7 — Digital
Digital (47, 65, 2452)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(765, 2452, F7, 18) (dual of [2452, 2387, 19]-code), using
- 43 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 28 times 0) [i] based on linear OA(761, 2405, F7, 18) (dual of [2405, 2344, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(761, 2401, F7, 18) (dual of [2401, 2340, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(757, 2401, F7, 17) (dual of [2401, 2344, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 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
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 43 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 28 times 0) [i] based on linear OA(761, 2405, F7, 18) (dual of [2405, 2344, 19]-code), using
(65−18, 65, 877169)-Net in Base 7 — Upper bound on s
There is no (47, 65, 877170)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 8 538403 183009 018571 810714 898038 303434 016241 298843 012765 > 765 [i]