Best Known (44, 44+18, s)-Nets in Base 7
(44, 44+18, 267)-Net over F7 — Constructive and digital
Digital (44, 62, 267)-net over F7, using
- 71 times duplication [i] based on digital (43, 61, 267)-net over F7, using
- net defined by OOA [i] based on linear OOA(761, 267, F7, 18, 18) (dual of [(267, 18), 4745, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(761, 2403, F7, 18) (dual of [2403, 2342, 19]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(761, 2405, F7, 18) (dual of [2405, 2344, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(761, 2403, F7, 18) (dual of [2403, 2342, 19]-code), using
- net defined by OOA [i] based on linear OOA(761, 267, F7, 18, 18) (dual of [(267, 18), 4745, 19]-NRT-code), using
(44, 44+18, 1880)-Net over F7 — Digital
Digital (44, 62, 1880)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(762, 1880, F7, 18) (dual of [1880, 1818, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(762, 2410, F7, 18) (dual of [2410, 2348, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [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(71, 9, F7, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(762, 2410, F7, 18) (dual of [2410, 2348, 19]-code), using
(44, 44+18, 458544)-Net in Base 7 — Upper bound on s
There is no (44, 62, 458545)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 24893 316586 302783 373687 313859 792746 931691 893229 293415 > 762 [i]