Best Known (27−8, 27, s)-Nets in Base 7
(27−8, 27, 602)-Net over F7 — Constructive and digital
Digital (19, 27, 602)-net over F7, using
- net defined by OOA [i] based on linear OOA(727, 602, F7, 8, 8) (dual of [(602, 8), 4789, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(727, 2408, F7, 8) (dual of [2408, 2381, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(727, 2409, F7, 8) (dual of [2409, 2382, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(717, 2401, F7, 5) (dual of [2401, 2384, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(72, 8, F7, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
- extended Reed–Solomon code RSe(6,7) [i]
- Hamming code H(2,7) [i]
- algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using the rational function field F7(x) [i]
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(727, 2409, F7, 8) (dual of [2409, 2382, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(727, 2408, F7, 8) (dual of [2408, 2381, 9]-code), using
(27−8, 27, 2289)-Net over F7 — Digital
Digital (19, 27, 2289)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(727, 2289, F7, 8) (dual of [2289, 2262, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(727, 2409, F7, 8) (dual of [2409, 2382, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(717, 2401, F7, 5) (dual of [2401, 2384, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(72, 8, F7, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,7)), using
- extended Reed–Solomon code RSe(6,7) [i]
- Hamming code H(2,7) [i]
- algebraic-geometric code AG(F, Q+1P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using the rational function field F7(x) [i]
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(727, 2409, F7, 8) (dual of [2409, 2382, 9]-code), using
(27−8, 27, 186770)-Net in Base 7 — Upper bound on s
There is no (19, 27, 186771)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 65713 059583 343592 792841 > 727 [i]